precsexlem11 - Metamath Proof Explorer

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Theorem precsexlem11 27903

Description: Lemma for surreal reciprocal. Show that the cut of the left and right sets is a multiplicative inverse for 𝐴. (Contributed by Scott Fenton, 15-Mar-2025.)

**Hypotheses**
Ref	Expression
precsexlem.1	⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
precsexlem.2	⊢ 𝐿 = (1^st ∘ 𝐹)
precsexlem.3	⊢ 𝑅 = (2^nd ∘ 𝐹)
precsexlem.4	⊢ (𝜑 → 𝐴 ∈ No )
precsexlem.5	⊢ (𝜑 → 0_s <s 𝐴)
precsexlem.6	⊢ (𝜑 → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
precsexlem.7	⊢ 𝑌 = (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))

**Assertion**
Ref	Expression
precsexlem11	⊢ (𝜑 → (𝐴 ·_s 𝑌) = 1_s )

Distinct variable groups: 𝐴,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥_𝑂,𝑥_𝐿,𝑥_𝑅,𝑦,𝑦_𝐿,𝑦_𝑅 𝐹,𝑙,𝑝 𝐿,𝑎,𝑙,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅 𝑅,𝑎,𝑙,𝑟,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅 𝜑,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅 𝐿,𝑟 𝜑,𝑟 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑝,𝑙,𝑥_𝑂) 𝑅(𝑥,𝑦,𝑝,𝑥_𝑂) 𝐹(𝑥,𝑦,𝑟,𝑎,𝑥_𝑂,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅) 𝐿(𝑥,𝑦,𝑝,𝑥_𝑂) 𝑌(𝑥,𝑦,𝑟,𝑝,𝑎,𝑙,𝑥_𝑂,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅)

Proof of Theorem precsexlem11 Dummy variables 𝑖 𝑗 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lltropt 27605	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
2		precsexlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
3		precsexlem.5	. . . . . . 7 ⊢ (𝜑 → 0_s <s 𝐴)
4	2, 3	0elleft 27642	. . . . . 6 ⊢ (𝜑 → 0_s ∈ ( L ‘𝐴))
5	4	snssd 4812	. . . . 5 ⊢ (𝜑 → { 0_s } ⊆ ( L ‘𝐴))
6		ssrab2 4077	. . . . . 6 ⊢ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ⊆ ( L ‘𝐴)
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ⊆ ( L ‘𝐴))
8	5, 7	unssd 4186	. . . 4 ⊢ (𝜑 → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ⊆ ( L ‘𝐴))
9		sssslt1 27534	. . . 4 ⊢ ((( L ‘𝐴) <<s ( R ‘𝐴) ∧ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ⊆ ( L ‘𝐴)) → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) <<s ( R ‘𝐴))
10	1, 8, 9	sylancr 586	. . 3 ⊢ (𝜑 → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) <<s ( R ‘𝐴))
11		precsexlem.1	. . . 4 ⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
12		precsexlem.2	. . . 4 ⊢ 𝐿 = (1^st ∘ 𝐹)
13		precsexlem.3	. . . 4 ⊢ 𝑅 = (2^nd ∘ 𝐹)
14		precsexlem.6	. . . 4 ⊢ (𝜑 → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
15	11, 12, 13, 2, 3, 14	precsexlem10 27902	. . 3 ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))
16	2, 3	cutpos 27659	. . 3 ⊢ (𝜑 → 𝐴 = (({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) \|s ( R ‘𝐴)))
17		precsexlem.7	. . . 4 ⊢ 𝑌 = (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))
18	17	a1i 11	. . 3 ⊢ (𝜑 → 𝑌 = (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)))
19	10, 15, 16, 18	mulsunif 27845	. 2 ⊢ (𝜑 → (𝐴 ·_s 𝑌) = (({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) \|s ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})))
20		0sno 27565	. . . . . . . . 9 ⊢ 0_s ∈ No
21	20	elexi 3493	. . . . . . . 8 ⊢ 0_s ∈ V
22	21	snid 4664	. . . . . . 7 ⊢ 0_s ∈ { 0_s }
23		elun1 4176	. . . . . . 7 ⊢ ( 0_s ∈ { 0_s } → 0_s ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}))
24	22, 23	ax-mp 5	. . . . . 6 ⊢ 0_s ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})
25		peano1 7883	. . . . . . . . 9 ⊢ ∅ ∈ ω
26	11, 12, 13	precsexlem1 27893	. . . . . . . . . 10 ⊢ (𝐿‘∅) = { 0_s }
27	22, 26	eleqtrri 2831	. . . . . . . . 9 ⊢ 0_s ∈ (𝐿‘∅)
28		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → (𝐿‘𝑏) = (𝐿‘∅))
29	28	eleq2d 2818	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ( 0_s ∈ (𝐿‘𝑏) ↔ 0_s ∈ (𝐿‘∅)))
30	29	rspcev 3612	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ 0_s ∈ (𝐿‘∅)) → ∃𝑏 ∈ ω 0_s ∈ (𝐿‘𝑏))
31	25, 27, 30	mp2an 689	. . . . . . . 8 ⊢ ∃𝑏 ∈ ω 0_s ∈ (𝐿‘𝑏)
32		eliun 5001	. . . . . . . 8 ⊢ ( 0_s ∈ ∪ 𝑏 ∈ ω (𝐿‘𝑏) ↔ ∃𝑏 ∈ ω 0_s ∈ (𝐿‘𝑏))
33	31, 32	mpbir 230	. . . . . . 7 ⊢ 0_s ∈ ∪ 𝑏 ∈ ω (𝐿‘𝑏)
34		fo1st 7999	. . . . . . . . . . 11 ⊢ 1^st :V–onto→V
35		fofun 6806	. . . . . . . . . . 11 ⊢ (1^st :V–onto→V → Fun 1^st )
36	34, 35	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 1^st
37		rdgfun 8420	. . . . . . . . . . 11 ⊢ Fun rec((𝑝 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
38	11	funeqi 6569	. . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ Fun rec((𝑝 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩))
39	37, 38	mpbir 230	. . . . . . . . . 10 ⊢ Fun 𝐹
40		funco 6588	. . . . . . . . . 10 ⊢ ((Fun 1^st ∧ Fun 𝐹) → Fun (1^st ∘ 𝐹))
41	36, 39, 40	mp2an 689	. . . . . . . . 9 ⊢ Fun (1^st ∘ 𝐹)
42	12	funeqi 6569	. . . . . . . . 9 ⊢ (Fun 𝐿 ↔ Fun (1^st ∘ 𝐹))
43	41, 42	mpbir 230	. . . . . . . 8 ⊢ Fun 𝐿
44		funiunfv 7250	. . . . . . . 8 ⊢ (Fun 𝐿 → ∪ 𝑏 ∈ ω (𝐿‘𝑏) = ∪ (𝐿 “ ω))
45	43, 44	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑏 ∈ ω (𝐿‘𝑏) = ∪ (𝐿 “ ω)
46	33, 45	eleqtri 2830	. . . . . 6 ⊢ 0_s ∈ ∪ (𝐿 “ ω)
47		addsrid 27687	. . . . . . . . . 10 ⊢ ( 0_s ∈ No → ( 0_s +_s 0_s ) = 0_s )
48	20, 47	ax-mp 5	. . . . . . . . 9 ⊢ ( 0_s +_s 0_s ) = 0_s
49		muls01 27808	. . . . . . . . . 10 ⊢ ( 0_s ∈ No → ( 0_s ·_s 0_s ) = 0_s )
50	20, 49	ax-mp 5	. . . . . . . . 9 ⊢ ( 0_s ·_s 0_s ) = 0_s
51	48, 50	oveq12i 7424	. . . . . . . 8 ⊢ (( 0_s +_s 0_s ) -_s ( 0_s ·_s 0_s )) = ( 0_s -_s 0_s )
52		subsid 27777	. . . . . . . . 9 ⊢ ( 0_s ∈ No → ( 0_s -_s 0_s ) = 0_s )
53	20, 52	ax-mp 5	. . . . . . . 8 ⊢ ( 0_s -_s 0_s ) = 0_s
54	51, 53	eqtr2i 2760	. . . . . . 7 ⊢ 0_s = (( 0_s +_s 0_s ) -_s ( 0_s ·_s 0_s ))
55	15	scutcld 27542	. . . . . . . . . . 11 ⊢ (𝜑 → (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)) ∈ No )
56	17, 55	eqeltrid 2836	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ No )
57		muls02 27837	. . . . . . . . . 10 ⊢ (𝑌 ∈ No → ( 0_s ·_s 𝑌) = 0_s )
58	56, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ( 0_s ·_s 𝑌) = 0_s )
59		muls01 27808	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴 ·_s 0_s ) = 0_s )
60	2, 59	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ·_s 0_s ) = 0_s )
61	58, 60	oveq12d 7430	. . . . . . . 8 ⊢ (𝜑 → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) = ( 0_s +_s 0_s ))
62	61	oveq1d 7427	. . . . . . 7 ⊢ (𝜑 → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) -_s ( 0_s ·_s 0_s )) = (( 0_s +_s 0_s ) -_s ( 0_s ·_s 0_s )))
63	54, 62	eqtr4id 2790	. . . . . 6 ⊢ (𝜑 → 0_s = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) -_s ( 0_s ·_s 0_s )))
64		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑐 = 0_s → (𝑐 ·_s 𝑌) = ( 0_s ·_s 𝑌))
65	64	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑐 = 0_s → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)))
66		oveq1 7419	. . . . . . . . 9 ⊢ (𝑐 = 0_s → (𝑐 ·_s 𝑑) = ( 0_s ·_s 𝑑))
67	65, 66	oveq12d 7430	. . . . . . . 8 ⊢ (𝑐 = 0_s → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)))
68	67	eqeq2d 2742	. . . . . . 7 ⊢ (𝑐 = 0_s → ( 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ 0_s = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑))))
69		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑑 = 0_s → (𝐴 ·_s 𝑑) = (𝐴 ·_s 0_s ))
70	69	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑑 = 0_s → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )))
71		oveq2 7420	. . . . . . . . 9 ⊢ (𝑑 = 0_s → ( 0_s ·_s 𝑑) = ( 0_s ·_s 0_s ))
72	70, 71	oveq12d 7430	. . . . . . . 8 ⊢ (𝑑 = 0_s → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) -_s ( 0_s ·_s 0_s )))
73	72	eqeq2d 2742	. . . . . . 7 ⊢ (𝑑 = 0_s → ( 0_s = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) ↔ 0_s = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) -_s ( 0_s ·_s 0_s ))))
74	68, 73	rspc2ev 3624	. . . . . 6 ⊢ (( 0_s ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 0_s ∈ ∪ (𝐿 “ ω) ∧ 0_s = ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 0_s )) -_s ( 0_s ·_s 0_s ))) → ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω) 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
75	24, 46, 63, 74	mp3an12i 1464	. . . . 5 ⊢ (𝜑 → ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω) 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
76		eqeq1 2735	. . . . . . 7 ⊢ (𝑏 = 0_s → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
77	76	2rexbidv 3218	. . . . . 6 ⊢ (𝑏 = 0_s → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω) 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
78	21, 77	elab 3668	. . . . 5 ⊢ ( 0_s ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω) 0_s = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
79	75, 78	sylibr 233	. . . 4 ⊢ (𝜑 → 0_s ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})
80		elun1 4176	. . . 4 ⊢ ( 0_s ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} → 0_s ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}))
81	79, 80	syl 17	. . 3 ⊢ (𝜑 → 0_s ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}))
82		eqid 2731	. . . . . . 7 ⊢ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
83	82	rnmpo 7545	. . . . . 6 ⊢ ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}
84		ssltex1 27525	. . . . . . . . 9 ⊢ (({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) <<s ( R ‘𝐴) → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∈ V)
85	10, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∈ V)
86		ssltex1 27525	. . . . . . . . 9 ⊢ (∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω) → ∪ (𝐿 “ ω) ∈ V)
87	15, 86	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ (𝐿 “ ω) ∈ V)
88		mpoexga 8068	. . . . . . . 8 ⊢ ((({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∈ V ∧ ∪ (𝐿 “ ω) ∈ V) → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
89	85, 87, 88	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
90		rnexg 7899	. . . . . . 7 ⊢ ((𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V → ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
91	89, 90	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
92	83, 91	eqeltrrid 2837	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∈ V)
93		eqid 2731	. . . . . . 7 ⊢ (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
94	93	rnmpo 7545	. . . . . 6 ⊢ ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}
95		fvex 6904	. . . . . . . 8 ⊢ ( R ‘𝐴) ∈ V
96		ssltex2 27526	. . . . . . . . 9 ⊢ (∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω) → ∪ (𝑅 “ ω) ∈ V)
97	15, 96	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ (𝑅 “ ω) ∈ V)
98		mpoexga 8068	. . . . . . . 8 ⊢ ((( R ‘𝐴) ∈ V ∧ ∪ (𝑅 “ ω) ∈ V) → (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
99	95, 97, 98	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
100		rnexg 7899	. . . . . . 7 ⊢ ((𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V → ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
101	99, 100	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
102	94, 101	eqeltrrid 2837	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∈ V)
103	92, 102	unexd 7745	. . . 4 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ∈ V)
104		snex 5431	. . . . 5 ⊢ { 1_s } ∈ V
105	104	a1i 11	. . . 4 ⊢ (𝜑 → { 1_s } ∈ V)
106		ssltss1 27527	. . . . . . . . . . . . . 14 ⊢ (({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) <<s ( R ‘𝐴) → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ⊆ No )
107	10, 106	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ⊆ No )
108	107	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})) → 𝑐 ∈ No )
109	108	adantrr 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑐 ∈ No )
110	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 ∈ No )
111	109, 110	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
112	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝐴 ∈ No )
113		ssltss1 27527	. . . . . . . . . . . . . 14 ⊢ (∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω) → ∪ (𝐿 “ ω) ⊆ No )
114	15, 113	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝐿 “ ω) ⊆ No )
115	114	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → 𝑑 ∈ No )
116	115	adantrl 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑑 ∈ No )
117	112, 116	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
118	111, 117	addscld 27703	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
119	109, 116	mulscld 27831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
120	118, 119	subscld 27775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∈ No )
121		eleq1 2820	. . . . . . . 8 ⊢ (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → (𝑏 ∈ No ↔ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∈ No ))
122	120, 121	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
123	122	rexlimdvva 3210	. . . . . 6 ⊢ (𝜑 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
124	123	abssdv 4065	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ⊆ No )
125		rightssno 27614	. . . . . . . . . . . . . 14 ⊢ ( R ‘𝐴) ⊆ No
126	125	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( R ‘𝐴) ⊆ No )
127	126	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝑐 ∈ No )
128	127	adantrr 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑐 ∈ No )
129	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 ∈ No )
130	128, 129	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
131	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝐴 ∈ No )
132		ssltss2 27528	. . . . . . . . . . . . . 14 ⊢ (∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω) → ∪ (𝑅 “ ω) ⊆ No )
133	15, 132	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝑅 “ ω) ⊆ No )
134	133	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → 𝑑 ∈ No )
135	134	adantrl 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑑 ∈ No )
136	131, 135	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
137	130, 136	addscld 27703	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
138	128, 135	mulscld 27831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
139	137, 138	subscld 27775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∈ No )
140	139, 121	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
141	140	rexlimdvva 3210	. . . . . 6 ⊢ (𝜑 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
142	141	abssdv 4065	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ⊆ No )
143	124, 142	unssd 4186	. . . 4 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ⊆ No )
144		1sno 27566	. . . . 5 ⊢ 1_s ∈ No
145		snssi 4811	. . . . 5 ⊢ ( 1_s ∈ No → { 1_s } ⊆ No )
146	144, 145	mp1i 13	. . . 4 ⊢ (𝜑 → { 1_s } ⊆ No )
147		elun 4148	. . . . . . . . 9 ⊢ (𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∨ 𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}))
148		vex 3477	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
149		eqeq1 2735	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑒 → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ 𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
150	149	2rexbidv 3218	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑒 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
151	148, 150	elab 3668	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
152	149	2rexbidv 3218	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑒 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
153	148, 152	elab 3668	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ↔ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
154	151, 153	orbi12i 912	. . . . . . . . 9 ⊢ ((𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∨ 𝑒 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
155	147, 154	bitri 275	. . . . . . . 8 ⊢ (𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
156		elun 4148	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ↔ (𝑐 ∈ { 0_s } ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}))
157		velsn 4644	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ { 0_s } ↔ 𝑐 = 0_s )
158	157	orbi1i 911	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ { 0_s } ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ↔ (𝑐 = 0_s ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}))
159	156, 158	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ↔ (𝑐 = 0_s ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}))
160	58	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ( 0_s ·_s 𝑌) = 0_s )
161	160	oveq1d 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = ( 0_s +_s (𝐴 ·_s 𝑑)))
162		muls02 27837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ No → ( 0_s ·_s 𝑑) = 0_s )
163	115, 162	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ( 0_s ·_s 𝑑) = 0_s )
164	161, 163	oveq12d 7430	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) = (( 0_s +_s (𝐴 ·_s 𝑑)) -_s 0_s ))
165	2	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → 𝐴 ∈ No )
166	165, 115	mulscld 27831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → (𝐴 ·_s 𝑑) ∈ No )
167		addslid 27691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ·_s 𝑑) ∈ No → ( 0_s +_s (𝐴 ·_s 𝑑)) = (𝐴 ·_s 𝑑))
168	166, 167	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ( 0_s +_s (𝐴 ·_s 𝑑)) = (𝐴 ·_s 𝑑))
169	168	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → (( 0_s +_s (𝐴 ·_s 𝑑)) -_s 0_s ) = ((𝐴 ·_s 𝑑) -_s 0_s ))
170		subsid1 27776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ·_s 𝑑) ∈ No → ((𝐴 ·_s 𝑑) -_s 0_s ) = (𝐴 ·_s 𝑑))
171	166, 170	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ((𝐴 ·_s 𝑑) -_s 0_s ) = (𝐴 ·_s 𝑑))
172	164, 169, 171	3eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) = (𝐴 ·_s 𝑑))
173		eliun 5001	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ ∃𝑖 ∈ ω 𝑑 ∈ (𝐿‘𝑖))
174		funiunfv 7250	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Fun 𝐿 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) = ∪ (𝐿 “ ω))
175	43, 174	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝑖 ∈ ω (𝐿‘𝑖) = ∪ (𝐿 “ ω)
176	175	eleq2i 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ 𝑑 ∈ ∪ (𝐿 “ ω))
177	173, 176	bitr3i 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑖 ∈ ω 𝑑 ∈ (𝐿‘𝑖) ↔ 𝑑 ∈ ∪ (𝐿 “ ω))
178	11, 12, 13, 2, 3, 14	precsexlem9 27901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (∀𝑑 ∈ (𝐿‘𝑖)(𝐴 ·_s 𝑑) <s 1_s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1_s <s (𝐴 ·_s 𝑐)))
179	178	simpld 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∀𝑑 ∈ (𝐿‘𝑖)(𝐴 ·_s 𝑑) <s 1_s )
180		rsp 3243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑑 ∈ (𝐿‘𝑖)(𝐴 ·_s 𝑑) <s 1_s → (𝑑 ∈ (𝐿‘𝑖) → (𝐴 ·_s 𝑑) <s 1_s ))
181	179, 180	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑑 ∈ (𝐿‘𝑖) → (𝐴 ·_s 𝑑) <s 1_s ))
182	181	rexlimdva 3154	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∃𝑖 ∈ ω 𝑑 ∈ (𝐿‘𝑖) → (𝐴 ·_s 𝑑) <s 1_s ))
183	177, 182	biimtrrid 242	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑑 ∈ ∪ (𝐿 “ ω) → (𝐴 ·_s 𝑑) <s 1_s ))
184	183	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → (𝐴 ·_s 𝑑) <s 1_s )
185	172, 184	eqbrtrd 5170	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝐿 “ ω)) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) <s 1_s )
186	185	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑑 ∈ ∪ (𝐿 “ ω) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) <s 1_s ))
187	67	breq1d 5158	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 0_s → ((((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ↔ ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) <s 1_s ))
188	187	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 0_s → ((𝑑 ∈ ∪ (𝐿 “ ω) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ) ↔ (𝑑 ∈ ∪ (𝐿 “ ω) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) <s 1_s )))
189	186, 188	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐 = 0_s → (𝑑 ∈ ∪ (𝐿 “ ω) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )))
190		scutcut 27540	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω) → ((∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)) ∈ No ∧ ∪ (𝐿 “ ω) <<s {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} ∧ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} <<s ∪ (𝑅 “ ω)))
191	15, 190	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)) ∈ No ∧ ∪ (𝐿 “ ω) <<s {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} ∧ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} <<s ∪ (𝑅 “ ω)))
192	191	simp3d 1143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} <<s ∪ (𝑅 “ ω))
193	192	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} <<s ∪ (𝑅 “ ω))
194		ovex 7445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)) ∈ V
195	194	snid 4664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω)) ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))}
196	17, 195	eqeltri 2828	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))}
197	196	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
198		peano2 7885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
199	198	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → suc 𝑖 ∈ ω)
200		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)
201		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥_𝐿 = 𝑐 → (𝑥_𝐿 -_s 𝐴) = (𝑐 -_s 𝐴))
202	201	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥_𝐿 = 𝑐 → ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿) = ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿))
203	202	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥_𝐿 = 𝑐 → ( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)))
204		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥_𝐿 = 𝑐 → 𝑥_𝐿 = 𝑐)
205	203, 204	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥_𝐿 = 𝑐 → (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐))
206	205	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥_𝐿 = 𝑐 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐)))
207		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦_𝐿 = 𝑑 → ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿) = ((𝑐 -_s 𝐴) ·_s 𝑑))
208	207	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦_𝐿 = 𝑑 → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
209	208	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦_𝐿 = 𝑑 → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐))
210	209	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦_𝐿 = 𝑑 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)))
211	206, 210	rspc2ev 3624	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ (𝐿‘𝑖) ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿))
212	200, 211	mp3an3 1449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ (𝐿‘𝑖)) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿))
213	212	ad2ant2l 743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿))
214		ovex 7445	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ V
215		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)))
216	215	2rexbidv 3218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) ↔ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)))
217	214, 216	elab 3668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ↔ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿))
218	213, 217	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)})
219		elun1 4176	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))
220		elun2 4177	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
221	218, 219, 220	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
222	11, 12, 13	precsexlem5 27897	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ ω → (𝑅‘suc 𝑖) = ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
223	222	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (𝑅‘suc 𝑖) = ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
224	221, 223	eleqtrrd 2835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘suc 𝑖))
225		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = suc 𝑖 → (𝑅‘𝑗) = (𝑅‘suc 𝑖))
226	225	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = suc 𝑖 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘suc 𝑖)))
227	226	rspcev 3612	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((suc 𝑖 ∈ ω ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘suc 𝑖)) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗))
228	199, 224, 227	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗))
229	228	rexlimdvaa 3155	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (∃𝑖 ∈ ω 𝑑 ∈ (𝐿‘𝑖) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗)))
230		eliun 5001	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗))
231		fo2nd 8000	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2^nd :V–onto→V
232		fofun 6806	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2^nd :V–onto→V → Fun 2^nd )
233	231, 232	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Fun 2^nd
234		funco 6588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 2^nd ∧ Fun 𝐹) → Fun (2^nd ∘ 𝐹))
235	233, 39, 234	mp2an 689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Fun (2^nd ∘ 𝐹)
236	13	funeqi 6569	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Fun 𝑅 ↔ Fun (2^nd ∘ 𝐹))
237	235, 236	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Fun 𝑅
238		funiunfv 7250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Fun 𝑅 → ∪ 𝑗 ∈ ω (𝑅‘𝑗) = ∪ (𝑅 “ ω))
239	237, 238	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑗 ∈ ω (𝑅‘𝑗) = ∪ (𝑅 “ ω)
240	239	eleq2i 2824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω))
241	230, 240	bitr3i 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω))
242	229, 177, 241	3imtr3g 295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝐿 “ ω) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω)))
243	242	impr 454	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω))
244	193, 197, 243	ssltsepcd 27533	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 <s (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐))
245	56	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 ∈ No )
246	144	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 1_s ∈ No )
247		leftssno 27613	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( L ‘𝐴) ⊆ No
248	6, 247	sstri 3991	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ⊆ No
249	248	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → 𝑐 ∈ No )
250	249	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → 𝑐 ∈ No )
251	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → 𝐴 ∈ No )
252	250, 251	subscld 27775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑐 -_s 𝐴) ∈ No )
253	252	adantrr 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 -_s 𝐴) ∈ No )
254	115	adantrl 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑑 ∈ No )
255	253, 254	mulscld 27831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) ∈ No )
256	246, 255	addscld 27703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ∈ No )
257	249	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑐 ∈ No )
258		breq2 5152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑐 → ( 0_s <s 𝑥 ↔ 0_s <s 𝑐))
259	258	elrab 3683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ↔ (𝑐 ∈ ( L ‘𝐴) ∧ 0_s <s 𝑐))
260	259	simprbi 496	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → 0_s <s 𝑐)
261	260	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 0_s <s 𝑐)
262	260	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → 0_s <s 𝑐)
263		breq2 5152	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥_𝑂 = 𝑐 → ( 0_s <s 𝑥_𝑂 ↔ 0_s <s 𝑐))
264		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥_𝑂 = 𝑐 → (𝑥_𝑂 ·_s 𝑦) = (𝑐 ·_s 𝑦))
265	264	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥_𝑂 = 𝑐 → ((𝑥_𝑂 ·_s 𝑦) = 1_s ↔ (𝑐 ·_s 𝑦) = 1_s ))
266	265	rexbidv 3177	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥_𝑂 = 𝑐 → (∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s ))
267	263, 266	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥_𝑂 = 𝑐 → (( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ) ↔ ( 0_s <s 𝑐 → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )))
268	14	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
269		ssun1 4172	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
270	6, 269	sstri 3991	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
271	270	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → 𝑐 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
272	271	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → 𝑐 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
273	267, 268, 272	rspcdva 3613	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → ( 0_s <s 𝑐 → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s ))
274	262, 273	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
275	274	adantrr 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
276	245, 256, 257, 261, 275	sltmuldiv2wd 27889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 ·_s 𝑌) <s ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ↔ 𝑌 <s (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)))
277	244, 276	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑌) <s ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
278	257, 254	mulscld 27831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
279	166	adantrl 713	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
280	246, 278, 279	addsubsassd 27788	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
281	2	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝐴 ∈ No )
282	257, 281, 254	subsdird 27854	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) = ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑)))
283	282	oveq2d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
284	280, 283	eqtr4d 2774	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
285	277, 284	breqtrrd 5176	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)))
286	56	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → 𝑌 ∈ No )
287	250, 286	mulscld 27831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑐 ·_s 𝑌) ∈ No )
288	287	adantrr 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
289	288, 279	addscld 27703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
290	289, 278, 246	sltsubaddd 27796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ↔ ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) <s ( 1_s +_s (𝑐 ·_s 𝑑))))
291	246, 278	addscld 27703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s (𝑐 ·_s 𝑑)) ∈ No )
292	288, 279, 291	sltaddsubd 27798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) <s ( 1_s +_s (𝑐 ·_s 𝑑)) ↔ (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑))))
293	290, 292	bitrd 279	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ↔ (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑))))
294	285, 293	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )
295	294	exp32 420	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → (𝑑 ∈ ∪ (𝐿 “ ω) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )))
296	189, 295	jaod 856	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑐 = 0_s ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝐿 “ ω) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )))
297	159, 296	biimtrid 241	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝐿 “ ω) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )))
298	297	imp32 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )
299		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → (𝑒 <s 1_s ↔ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ))
300	298, 299	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑒 <s 1_s ))
301	300	rexlimdvva 3210	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑒 <s 1_s ))
302	192	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))} <<s ∪ (𝑅 “ ω))
303	196	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
304	198	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → suc 𝑖 ∈ ω)
305		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥_𝑅 = 𝑐 → (𝑥_𝑅 -_s 𝐴) = (𝑐 -_s 𝐴))
306	305	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥_𝑅 = 𝑐 → ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅) = ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅))
307	306	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥_𝑅 = 𝑐 → ( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)))
308		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥_𝑅 = 𝑐 → 𝑥_𝑅 = 𝑐)
309	307, 308	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥_𝑅 = 𝑐 → (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐))
310	309	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥_𝑅 = 𝑐 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐)))
311		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦_𝑅 = 𝑑 → ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅) = ((𝑐 -_s 𝐴) ·_s 𝑑))
312	311	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦_𝑅 = 𝑑 → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
313	312	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦_𝑅 = 𝑑 → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐))
314	313	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦_𝑅 = 𝑑 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)))
315	310, 314	rspc2ev 3624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ (𝑅‘𝑖) ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅))
316	200, 315	mp3an3 1449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ (𝑅‘𝑖)) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅))
317	316	ad2ant2l 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅))
318		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)))
319	318	2rexbidv 3218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) ↔ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)))
320	214, 319	elab 3668	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)} ↔ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅))
321	317, 320	sylibr 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})
322		elun2 4177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)} → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))
323	321, 322, 220	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
324	222	ad2antrl 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (𝑅‘suc 𝑖) = ((𝑅‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
325	323, 324	eleqtrrd 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘suc 𝑖))
326	304, 325, 227	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗))
327	326	rexlimdvaa 3155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (∃𝑖 ∈ ω 𝑑 ∈ (𝑅‘𝑖) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝑅‘𝑗)))
328		eliun 5001	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ ∪ 𝑖 ∈ ω (𝑅‘𝑖) ↔ ∃𝑖 ∈ ω 𝑑 ∈ (𝑅‘𝑖))
329		funiunfv 7250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑅 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) = ∪ (𝑅 “ ω))
330	237, 329	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝑖 ∈ ω (𝑅‘𝑖) = ∪ (𝑅 “ ω)
331	330	eleq2i 2824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ ∪ 𝑖 ∈ ω (𝑅‘𝑖) ↔ 𝑑 ∈ ∪ (𝑅 “ ω))
332	328, 331	bitr3i 277	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑖 ∈ ω 𝑑 ∈ (𝑅‘𝑖) ↔ 𝑑 ∈ ∪ (𝑅 “ ω))
333	327, 332, 241	3imtr3g 295	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (𝑑 ∈ ∪ (𝑅 “ ω) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω)))
334	333	impr 454	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝑅 “ ω))
335	302, 303, 334	ssltsepcd 27533	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 <s (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐))
336	144	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 1_s ∈ No )
337	2	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
338	127, 337	subscld 27775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (𝑐 -_s 𝐴) ∈ No )
339	338	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 -_s 𝐴) ∈ No )
340	339, 135	mulscld 27831	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) ∈ No )
341	336, 340	addscld 27703	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ∈ No )
342	20	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 0_s ∈ No )
343	3	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 0_s <s 𝐴)
344		breq2 5152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥_𝑂 = 𝑐 → (𝐴 <s 𝑥_𝑂 ↔ 𝐴 <s 𝑐))
345		rightval 27597	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( R ‘𝐴) = {𝑥_𝑂 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥_𝑂}
346	344, 345	elrab2 3686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ ( R ‘𝐴) ↔ (𝑐 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑐))
347	346	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ ( R ‘𝐴) → 𝐴 <s 𝑐)
348	347	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑐)
349	342, 337, 127, 343, 348	slttrd 27499	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 0_s <s 𝑐)
350	349	adantrr 714	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 0_s <s 𝑐)
351	14	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
352		elun2 4177	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ ( R ‘𝐴) → 𝑐 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
353	352	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → 𝑐 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
354	267, 351, 353	rspcdva 3613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → ( 0_s <s 𝑐 → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s ))
355	349, 354	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
356	355	adantrr 714	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
357	129, 341, 128, 350, 356	sltmuldiv2wd 27889	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 ·_s 𝑌) <s ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ↔ 𝑌 <s (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)))
358	335, 357	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑌) <s ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
359	336, 138, 136	addsubsassd 27788	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
360	128, 131, 135	subsdird 27854	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) = ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑)))
361	360	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
362	359, 361	eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
363	358, 362	breqtrrd 5176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)))
364	137, 138, 336	sltsubaddd 27796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ↔ ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) <s ( 1_s +_s (𝑐 ·_s 𝑑))))
365	336, 138	addscld 27703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s (𝑐 ·_s 𝑑)) ∈ No )
366	130, 136, 365	sltaddsubd 27798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) <s ( 1_s +_s (𝑐 ·_s 𝑑)) ↔ (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑))))
367	364, 366	bitrd 279	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s ↔ (𝑐 ·_s 𝑌) <s (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑))))
368	363, 367	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) <s 1_s )
369	368, 299	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑒 <s 1_s ))
370	369	rexlimdvva 3210	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑒 <s 1_s ))
371	301, 370	jaod 856	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑒 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) → 𝑒 <s 1_s ))
372	155, 371	biimtrid 241	. . . . . . 7 ⊢ (𝜑 → (𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 𝑒 <s 1_s ))
373	372	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})) → 𝑒 <s 1_s )
374		velsn 4644	. . . . . . 7 ⊢ (𝑓 ∈ { 1_s } ↔ 𝑓 = 1_s )
375		breq2 5152	. . . . . . 7 ⊢ (𝑓 = 1_s → (𝑒 <s 𝑓 ↔ 𝑒 <s 1_s ))
376	374, 375	sylbi 216	. . . . . 6 ⊢ (𝑓 ∈ { 1_s } → (𝑒 <s 𝑓 ↔ 𝑒 <s 1_s ))
377	373, 376	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})) → (𝑓 ∈ { 1_s } → 𝑒 <s 𝑓))
378	377	3impia 1116	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ∧ 𝑓 ∈ { 1_s }) → 𝑒 <s 𝑓)
379	103, 105, 143, 146, 378	ssltd 27530	. . 3 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) <<s { 1_s })
380		eqid 2731	. . . . . . 7 ⊢ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
381	380	rnmpo 7545	. . . . . 6 ⊢ ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}
382		mpoexga 8068	. . . . . . . 8 ⊢ ((({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∈ V ∧ ∪ (𝑅 “ ω) ∈ V) → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
383	85, 97, 382	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
384		rnexg 7899	. . . . . . 7 ⊢ ((𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V → ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
385	383, 384	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}), 𝑑 ∈ ∪ (𝑅 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
386	381, 385	eqeltrrid 2837	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∈ V)
387		eqid 2731	. . . . . . 7 ⊢ (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
388	387	rnmpo 7545	. . . . . 6 ⊢ ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) = {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}
389		mpoexga 8068	. . . . . . . 8 ⊢ ((( R ‘𝐴) ∈ V ∧ ∪ (𝐿 “ ω) ∈ V) → (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
390	95, 87, 389	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
391		rnexg 7899	. . . . . . 7 ⊢ ((𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V → ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
392	390, 391	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑐 ∈ ( R ‘𝐴), 𝑑 ∈ ∪ (𝐿 “ ω) ↦ (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ∈ V)
393	388, 392	eqeltrrid 2837	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∈ V)
394	386, 393	unexd 7745	. . . 4 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ∈ V)
395	108	adantrr 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑐 ∈ No )
396	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 ∈ No )
397	395, 396	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
398	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝐴 ∈ No )
399	134	adantrl 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑑 ∈ No )
400	398, 399	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
401	397, 400	addscld 27703	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
402	395, 399	mulscld 27831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
403	401, 402	subscld 27775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∈ No )
404	403, 121	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
405	404	rexlimdvva 3210	. . . . . 6 ⊢ (𝜑 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
406	405	abssdv 4065	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ⊆ No )
407	127	adantrr 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑐 ∈ No )
408	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 ∈ No )
409	407, 408	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
410	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝐴 ∈ No )
411	115	adantrl 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑑 ∈ No )
412	410, 411	mulscld 27831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
413	409, 412	addscld 27703	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
414	407, 411	mulscld 27831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
415	413, 414	subscld 27775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∈ No )
416	415, 121	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
417	416	rexlimdvva 3210	. . . . . 6 ⊢ (𝜑 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 𝑏 ∈ No ))
418	417	abssdv 4065	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ⊆ No )
419	406, 418	unssd 4186	. . . 4 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ⊆ No )
420		elun 4148	. . . . . . . 8 ⊢ (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∨ 𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}))
421		vex 3477	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
422		eqeq1 2735	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ 𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
423	422	2rexbidv 3218	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
424	421, 423	elab 3668	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ↔ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
425	422	2rexbidv 3218	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
426	421, 425	elab 3668	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ↔ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
427	424, 426	orbi12i 912	. . . . . . . 8 ⊢ ((𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∨ 𝑓 ∈ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
428	420, 427	bitri 275	. . . . . . 7 ⊢ (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) ↔ (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
429		eliun 5001	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ ∃𝑗 ∈ ω 𝑑 ∈ (𝑅‘𝑗))
430	239	eleq2i 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ 𝑑 ∈ ∪ (𝑅 “ ω))
431	429, 430	bitr3i 277	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑗 ∈ ω 𝑑 ∈ (𝑅‘𝑗) ↔ 𝑑 ∈ ∪ (𝑅 “ ω))
432	11, 12, 13, 2, 3, 14	precsexlem9 27901	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (∀𝑐 ∈ (𝐿‘𝑗)(𝐴 ·_s 𝑐) <s 1_s ∧ ∀𝑑 ∈ (𝑅‘𝑗) 1_s <s (𝐴 ·_s 𝑑)))
433		rsp 3243	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑑 ∈ (𝑅‘𝑗) 1_s <s (𝐴 ·_s 𝑑) → (𝑑 ∈ (𝑅‘𝑗) → 1_s <s (𝐴 ·_s 𝑑)))
434	432, 433	simpl2im 503	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑑 ∈ (𝑅‘𝑗) → 1_s <s (𝐴 ·_s 𝑑)))
435	434	rexlimdva 3154	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (∃𝑗 ∈ ω 𝑑 ∈ (𝑅‘𝑗) → 1_s <s (𝐴 ·_s 𝑑)))
436	431, 435	biimtrrid 242	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (𝐴 ·_s 𝑑)))
437	436	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → 1_s <s (𝐴 ·_s 𝑑))
438	56	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → 𝑌 ∈ No )
439	57	oveq1d 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 ∈ No → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = ( 0_s +_s (𝐴 ·_s 𝑑)))
440	438, 439	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = ( 0_s +_s (𝐴 ·_s 𝑑)))
441	2	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → 𝐴 ∈ No )
442	441, 134	mulscld 27831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → (𝐴 ·_s 𝑑) ∈ No )
443	442, 167	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → ( 0_s +_s (𝐴 ·_s 𝑑)) = (𝐴 ·_s 𝑑))
444	440, 443	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → (( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) = (𝐴 ·_s 𝑑))
445	134, 162	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → ( 0_s ·_s 𝑑) = 0_s )
446	444, 445	oveq12d 7430	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) = ((𝐴 ·_s 𝑑) -_s 0_s ))
447	442, 170	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → ((𝐴 ·_s 𝑑) -_s 0_s ) = (𝐴 ·_s 𝑑))
448	446, 447	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)) = (𝐴 ·_s 𝑑))
449	437, 448	breqtrrd 5176	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ ∪ (𝑅 “ ω)) → 1_s <s ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)))
450	449	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑))))
451	67	breq2d 5160	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 0_s → ( 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ↔ 1_s <s ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑))))
452	451	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 0_s → ((𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) ↔ (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s ((( 0_s ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s ( 0_s ·_s 𝑑)))))
453	450, 452	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑐 = 0_s → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))))
454	144	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 1_s ∈ No )
455	249	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑐 ∈ No )
456	134	adantrl 713	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑑 ∈ No )
457	455, 456	mulscld 27831	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑑) ∈ No )
458	442	adantrl 713	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝐴 ·_s 𝑑) ∈ No )
459	454, 457, 458	addsubsassd 27788	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
460	2	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝐴 ∈ No )
461	455, 460, 456	subsdird 27854	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) = ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑)))
462	461	oveq2d 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
463	459, 462	eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
464	191	simp2d 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
465	464	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ∪ (𝐿 “ ω) <<s {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
466	198	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → suc 𝑖 ∈ ω)
467	201	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥_𝐿 = 𝑐 → ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅) = ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅))
468	467	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥_𝐿 = 𝑐 → ( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)))
469	468, 204	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥_𝐿 = 𝑐 → (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐))
470	469	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥_𝐿 = 𝑐 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑐)))
471	470, 314	rspc2ev 3624	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ (𝑅‘𝑖) ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿))
472	200, 471	mp3an3 1449	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ (𝑅‘𝑖)) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿))
473	472	ad2ant2l 743	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿))
474		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)))
475	474	2rexbidv 3218	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) ↔ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)))
476	214, 475	elab 3668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)} ↔ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿))
477	473, 476	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})
478		elun2 4177	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)} → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)}))
479		elun2 4177	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)}) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
480	477, 478, 479	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
481	11, 12, 13	precsexlem4 27896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ ω → (𝐿‘suc 𝑖) = ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
482	481	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (𝐿‘suc 𝑖) = ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
483	480, 482	eleqtrrd 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘suc 𝑖))
484		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = suc 𝑖 → (𝐿‘𝑗) = (𝐿‘suc 𝑖))
485	484	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = suc 𝑖 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘suc 𝑖)))
486	485	rspcev 3612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((suc 𝑖 ∈ ω ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘suc 𝑖)) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗))
487	466, 483, 486	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝑅‘𝑖))) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗))
488	487	rexlimdvaa 3155	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (∃𝑖 ∈ ω 𝑑 ∈ (𝑅‘𝑖) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗)))
489		eliun 5001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ 𝑗 ∈ ω (𝐿‘𝑗) ↔ ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗))
490		funiunfv 7250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Fun 𝐿 → ∪ 𝑗 ∈ ω (𝐿‘𝑗) = ∪ (𝐿 “ ω))
491	43, 490	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ 𝑗 ∈ ω (𝐿‘𝑗) = ∪ (𝐿 “ ω)
492	491	eleq2i 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ 𝑗 ∈ ω (𝐿‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω))
493	489, 492	bitr3i 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω))
494	488, 332, 493	3imtr3g 295	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝑅 “ ω) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω)))
495	494	impr 454	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω))
496	196	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
497	465, 495, 496	ssltsepcd 27533	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) <s 𝑌)
498	252	adantrr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 -_s 𝐴) ∈ No )
499	498, 456	mulscld 27831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) ∈ No )
500	454, 499	addscld 27703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ∈ No )
501	56	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 𝑌 ∈ No )
502	260	ad2antrl 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 0_s <s 𝑐)
503	274	adantrr 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
504	500, 501, 455, 502, 503	sltdivmulwd 27886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) <s 𝑌 ↔ ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) <s (𝑐 ·_s 𝑌)))
505	497, 504	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) <s (𝑐 ·_s 𝑌))
506	463, 505	eqbrtrd 5170	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌))
507	454, 457	addscld 27703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ( 1_s +_s (𝑐 ·_s 𝑑)) ∈ No )
508	287	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑐 ·_s 𝑌) ∈ No )
509	507, 458, 508	sltsubaddd 27796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌) ↔ ( 1_s +_s (𝑐 ·_s 𝑑)) <s ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑))))
510	508, 458	addscld 27703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ∈ No )
511	454, 457, 510	sltaddsubd 27798	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) <s ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ↔ 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
512	509, 511	bitrd 279	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → ((( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌) ↔ 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
513	506, 512	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
514	513	exp32 420	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))))
515	453, 514	jaod 856	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑐 = 0_s ∨ 𝑐 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))))
516	159, 515	biimtrid 241	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) → (𝑑 ∈ ∪ (𝑅 “ ω) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))))
517	516	imp32 418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
518		breq2 5152	. . . . . . . . . 10 ⊢ (𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → ( 1_s <s 𝑓 ↔ 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
519	517, 518	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}) ∧ 𝑑 ∈ ∪ (𝑅 “ ω))) → (𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 1_s <s 𝑓))
520	519	rexlimdvva 3210	. . . . . . . 8 ⊢ (𝜑 → (∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 1_s <s 𝑓))
521	144	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 1_s ∈ No )
522	521, 414, 412	addsubsassd 27788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
523	407, 410, 411	subsdird 27854	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) = ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑)))
524	523	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) = ( 1_s +_s ((𝑐 ·_s 𝑑) -_s (𝐴 ·_s 𝑑))))
525	522, 524	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)))
526	464	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ∪ (𝐿 “ ω) <<s {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
527	198	ad2antrl 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → suc 𝑖 ∈ ω)
528	305	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥_𝑅 = 𝑐 → ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿) = ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿))
529	528	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥_𝑅 = 𝑐 → ( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) = ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)))
530	529, 308	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥_𝑅 = 𝑐 → (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐))
531	530	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥_𝑅 = 𝑐 → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑐)))
532	531, 210	rspc2ev 3624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ (𝐿‘𝑖) ∧ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐)) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅))
533	200, 532	mp3an3 1449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ (𝐿‘𝑖)) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅))
534	533	ad2ant2l 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅))
535		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) ↔ (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)))
536	535	2rexbidv 3218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) → (∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) ↔ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)))
537	214, 536	elab 3668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ↔ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)(( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅))
538	534, 537	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)})
539		elun1 4176	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)}))
540	538, 539, 479	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
541	481	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (𝐿‘suc 𝑖) = ((𝐿‘𝑖) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑖)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
542	540, 541	eleqtrrd 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘suc 𝑖))
543	527, 542, 486	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) ∧ (𝑖 ∈ ω ∧ 𝑑 ∈ (𝐿‘𝑖))) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗))
544	543	rexlimdvaa 3155	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (∃𝑖 ∈ ω 𝑑 ∈ (𝐿‘𝑖) → ∃𝑗 ∈ ω (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ (𝐿‘𝑗)))
545	544, 177, 493	3imtr3g 295	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ( R ‘𝐴)) → (𝑑 ∈ ∪ (𝐿 “ ω) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω)))
546	545	impr 454	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) ∈ ∪ (𝐿 “ ω))
547	196	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 𝑌 ∈ {(∪ (𝐿 “ ω) \|s ∪ (𝑅 “ ω))})
548	526, 546, 547	ssltsepcd 27533	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) <s 𝑌)
549	338	adantrr 714	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑐 -_s 𝐴) ∈ No )
550	549, 411	mulscld 27831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((𝑐 -_s 𝐴) ·_s 𝑑) ∈ No )
551	521, 550	addscld 27703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) ∈ No )
552	349	adantrr 714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 0_s <s 𝑐)
553	355	adantrr 714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ∃𝑦 ∈ No (𝑐 ·_s 𝑦) = 1_s )
554	551, 408, 407, 552, 553	sltdivmulwd 27886	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) /_su 𝑐) <s 𝑌 ↔ ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) <s (𝑐 ·_s 𝑌)))
555	548, 554	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s ((𝑐 -_s 𝐴) ·_s 𝑑)) <s (𝑐 ·_s 𝑌))
556	525, 555	eqbrtrd 5170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌))
557	521, 414	addscld 27703	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ( 1_s +_s (𝑐 ·_s 𝑑)) ∈ No )
558	557, 412, 409	sltsubaddd 27796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌) ↔ ( 1_s +_s (𝑐 ·_s 𝑑)) <s ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑))))
559	521, 414, 413	sltaddsubd 27798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (( 1_s +_s (𝑐 ·_s 𝑑)) <s ((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) ↔ 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
560	558, 559	bitrd 279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → ((( 1_s +_s (𝑐 ·_s 𝑑)) -_s (𝐴 ·_s 𝑑)) <s (𝑐 ·_s 𝑌) ↔ 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))))
561	556, 560	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → 1_s <s (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)))
562	561, 518	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ( R ‘𝐴) ∧ 𝑑 ∈ ∪ (𝐿 “ ω))) → (𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 1_s <s 𝑓))
563	562	rexlimdvva 3210	. . . . . . . 8 ⊢ (𝜑 → (∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) → 1_s <s 𝑓))
564	520, 563	jaod 856	. . . . . . 7 ⊢ (𝜑 → ((∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑)) ∨ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑓 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))) → 1_s <s 𝑓))
565	428, 564	biimtrid 241	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 1_s <s 𝑓))
566		velsn 4644	. . . . . . 7 ⊢ (𝑒 ∈ { 1_s } ↔ 𝑒 = 1_s )
567		breq1 5151	. . . . . . . 8 ⊢ (𝑒 = 1_s → (𝑒 <s 𝑓 ↔ 1_s <s 𝑓))
568	567	imbi2d 340	. . . . . . 7 ⊢ (𝑒 = 1_s → ((𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 𝑒 <s 𝑓) ↔ (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 1_s <s 𝑓)))
569	566, 568	sylbi 216	. . . . . 6 ⊢ (𝑒 ∈ { 1_s } → ((𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 𝑒 <s 𝑓) ↔ (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 1_s <s 𝑓)))
570	565, 569	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝑒 ∈ { 1_s } → (𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) → 𝑒 <s 𝑓)))
571	570	3imp 1110	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ { 1_s } ∧ 𝑓 ∈ ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})) → 𝑒 <s 𝑓)
572	105, 394, 146, 419, 571	ssltd 27530	. . 3 ⊢ (𝜑 → { 1_s } <<s ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}))
573	81, 379, 572	cuteq1 27572	. 2 ⊢ (𝜑 → (({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))}) \|s ({𝑏 ∣ ∃𝑐 ∈ ({ 0_s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})∃𝑑 ∈ ∪ (𝑅 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))} ∪ {𝑏 ∣ ∃𝑐 ∈ ( R ‘𝐴)∃𝑑 ∈ ∪ (𝐿 “ ω)𝑏 = (((𝑐 ·_s 𝑌) +_s (𝐴 ·_s 𝑑)) -_s (𝑐 ·_s 𝑑))})) = 1_s )
574	19, 573	eqtrd 2771	1 ⊢ (𝜑 → (𝐴 ·_s 𝑌) = 1_s )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ⦋csb 3893 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 ⟨cop 4634 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 “ cima 5679 ∘ ccom 5680 suc csuc 6366 Fun wfun 6537 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ωcom 7859 1^st c1st 7977 2^nd c2nd 7978 reccrdg 8413 No csur 27380 <s cslt 27381 bday cbday 27382 <<s csslt 27519 |s cscut 27521 0_s c0s 27561 1_s c1s 27562 O cold 27576 L cleft 27578 R cright 27579 +_s cadds 27682 -_s csubs 27735 ·_s cmuls 27802 /_su cdivs 27875

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-dc 10445

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-nadd 8669 df-no 27383 df-slt 27384 df-bday 27385 df-sle 27485 df-sslt 27520 df-scut 27522 df-0s 27563 df-1s 27564 df-made 27580 df-old 27581 df-left 27583 df-right 27584 df-norec 27661 df-norec2 27672 df-adds 27683 df-negs 27736 df-subs 27737 df-muls 27803 df-divs 27876

This theorem is referenced by: precsex 27904

W3C validator