Theorem precsexlem10 28228

Description: Lemma for surreal reciprocal. Show that the union of the left sets is less than the union of the right sets. Note that this is the first theorem in the surreal numbers to require the axiom of infinity. (Contributed by Scott Fenton, 15-Mar-2025.)

Hypotheses

Ref	Expression
precsexlem.1	⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
precsexlem.2	⊢ 𝐿 = (1st ∘ 𝐹)
precsexlem.3	⊢ 𝑅 = (2nd ∘ 𝐹)
precsexlem.4	⊢ (𝜑 → 𝐴 ∈ No )
precsexlem.5	⊢ (𝜑 → 0s <s 𝐴)
precsexlem.6	⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))

Assertion

Ref	Expression
precsexlem10	⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))

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)

Proof of Theorem precsexlem10

Dummy variables 𝑖 𝑗 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 7953	. . . . . . . 8 ⊢ 1st :V–onto→V
2		fofun 6743	. . . . . . . 8 ⊢ (1st :V–onto→V → Fun 1st )
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ Fun 1st
4		rdgfun 8349	. . . . . . . 8 ⊢ Fun rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
5		precsexlem.1	. . . . . . . . 9 ⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
6	5	funeqi 6509	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ Fun rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩))
7	4, 6	mpbir 233	. . . . . . 7 ⊢ Fun 𝐹
8		funco 6528	. . . . . . 7 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
9	3, 7, 8	mp2an 699	. . . . . 6 ⊢ Fun (1st ∘ 𝐹)
10		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1st ∘ 𝐹)
11	10	funeqi 6509	. . . . . 6 ⊢ (Fun 𝐿 ↔ Fun (1st ∘ 𝐹))
12	9, 11	mpbir 233	. . . . 5 ⊢ Fun 𝐿
13		dcomex 10365	. . . . . 6 ⊢ ω ∈ V
14	13	funimaex 6576	. . . . 5 ⊢ (Fun 𝐿 → (𝐿 “ ω) ∈ V)
15	12, 14	ax-mp 5	. . . 4 ⊢ (𝐿 “ ω) ∈ V
16	15	uniex 7687	. . 3 ⊢ ∪ (𝐿 “ ω) ∈ V
17	16	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ∈ V)
18		fo2nd 7954	. . . . . . . 8 ⊢ 2nd :V–onto→V
19		fofun 6743	. . . . . . . 8 ⊢ (2nd :V–onto→V → Fun 2nd )
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ Fun 2nd
21		funco 6528	. . . . . . 7 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
22	20, 7, 21	mp2an 699	. . . . . 6 ⊢ Fun (2nd ∘ 𝐹)
23		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2nd ∘ 𝐹)
24	23	funeqi 6509	. . . . . 6 ⊢ (Fun 𝑅 ↔ Fun (2nd ∘ 𝐹))
25	22, 24	mpbir 233	. . . . 5 ⊢ Fun 𝑅
26	13	funimaex 6576	. . . . 5 ⊢ (Fun 𝑅 → (𝑅 “ ω) ∈ V)
27	25, 26	ax-mp 5	. . . 4 ⊢ (𝑅 “ ω) ∈ V
28	27	uniex 7687	. . 3 ⊢ ∪ (𝑅 “ ω) ∈ V
29	28	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ∈ V)
30		funiunfv 7195	. . . 4 ⊢ (Fun 𝐿 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) = ∪ (𝐿 “ ω))
31	12, 30	ax-mp 5	. . 3 ⊢ ∪ 𝑖 ∈ ω (𝐿‘𝑖) = ∪ (𝐿 “ ω)
32		precsexlem.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
33		precsexlem.5	. . . . . 6 ⊢ (𝜑 → 0s <s 𝐴)
34		precsexlem.6	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
35	5, 10, 23, 32, 33, 34	precsexlem8 28226	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ))
36	35	simpld 496	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐿‘𝑖) ⊆ No )
37	36	iunssd 4982	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) ⊆ No )
38	31, 37	eqsstrrid 3955	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ⊆ No )
39		funiunfv 7195	. . . 4 ⊢ (Fun 𝑅 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) = ∪ (𝑅 “ ω))
40	25, 39	ax-mp 5	. . 3 ⊢ ∪ 𝑖 ∈ ω (𝑅‘𝑖) = ∪ (𝑅 “ ω)
41	35	simprd 497	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑅‘𝑖) ⊆ No )
42	41	iunssd 4982	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) ⊆ No )
43	40, 42	eqsstrrid 3955	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ⊆ No )
44	31	eleq2i 2833	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ 𝑏 ∈ ∪ (𝐿 “ ω))
45		eliun 4927	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
46	44, 45	bitr3i 279	. . . . . 6 ⊢ (𝑏 ∈ ∪ (𝐿 “ ω) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
47		funiunfv 7195	. . . . . . . . 9 ⊢ (Fun 𝑅 → ∪ 𝑗 ∈ ω (𝑅‘𝑗) = ∪ (𝑅 “ ω))
48	25, 47	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ω (𝑅‘𝑗) = ∪ (𝑅 “ ω)
49	48	eleq2i 2833	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ 𝑐 ∈ ∪ (𝑅 “ ω))
50		eliun 4927	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
51	49, 50	bitr3i 279	. . . . . 6 ⊢ (𝑐 ∈ ∪ (𝑅 “ ω) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
52	46, 51	anbi12i 635	. . . . 5 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ (∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖) ∧ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗)))
53		reeanv 3213	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) ↔ (∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖) ∧ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗)))
54	52, 53	bitr4i 280	. . . 4 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)))
55		omun 7831	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖 ∪ 𝑗) ∈ ω)
56		ssun1 4109	. . . . . . . . . 10 ⊢ 𝑖 ⊆ (𝑖 ∪ 𝑗)
57	5, 10, 23	precsexlem6 28224	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑖 ⊆ (𝑖 ∪ 𝑗)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
58	56, 57	mp3an3 1459	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
59	55, 58	syldan 598	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
60	59	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
61	60	sseld 3915	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑏 ∈ (𝐿‘𝑖) → 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))))
62		simpr 486	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
63		ssun2 4110	. . . . . . . . . 10 ⊢ 𝑗 ⊆ (𝑖 ∪ 𝑗)
64	5, 10, 23	precsexlem7 28225	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑗 ⊆ (𝑖 ∪ 𝑗)) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
65	63, 64	mp3an3 1459	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
66	62, 55, 65	syl2anc 591	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
67	66	sseld 3915	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
68	67	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
69	32	ad2antrr 733	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝐴 ∈ No )
70	5, 10, 23, 32, 33, 34	precsexlem8 28226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝐿‘(𝑖 ∪ 𝑗)) ⊆ No ∧ (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No ))
71	70	simpld 496	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘(𝑖 ∪ 𝑗)) ⊆ No )
72	71	sselda 3916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))) → 𝑏 ∈ No )
73	72	adantrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 ∈ No )
74	69, 73	mulscld 28147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑏) ∈ No )
75	70	simprd 497	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No )
76	75	sselda 3916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑐 ∈ No )
77	76	adantrl 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑐 ∈ No )
78	69, 77	mulscld 28147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑐) ∈ No )
79	74, 78	jca 517	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → ((𝐴 ·s 𝑏) ∈ No ∧ (𝐴 ·s 𝑐) ∈ No ))
80	5, 10, 23, 32, 33, 34	precsexlem9 28227	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐)))
81	80	simpld 496	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s )
82		rsp 3229	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s → (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) → (𝐴 ·s 𝑏) <s 1s ))
83	81, 82	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) → (𝐴 ·s 𝑏) <s 1s ))
84	80	simprd 497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐))
85		rsp 3229	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐) → (𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) → 1s <s (𝐴 ·s 𝑐)))
86	84, 85	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) → 1s <s (𝐴 ·s 𝑐)))
87	83, 86	anim12d 616	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → ((𝐴 ·s 𝑏) <s 1s ∧ 1s <s (𝐴 ·s 𝑐))))
88	87	imp 408	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → ((𝐴 ·s 𝑏) <s 1s ∧ 1s <s (𝐴 ·s 𝑐)))
89		1no 27822	. . . . . . . . . . 11 ⊢ 1s ∈ No
90		ltstr 27731	. . . . . . . . . . 11 ⊢ (((𝐴 ·s 𝑏) ∈ No ∧ 1s ∈ No ∧ (𝐴 ·s 𝑐) ∈ No ) → (((𝐴 ·s 𝑏) <s 1s ∧ 1s <s (𝐴 ·s 𝑐)) → (𝐴 ·s 𝑏) <s (𝐴 ·s 𝑐)))
91	89, 90	mp3an2 1458	. . . . . . . . . 10 ⊢ (((𝐴 ·s 𝑏) ∈ No ∧ (𝐴 ·s 𝑐) ∈ No ) → (((𝐴 ·s 𝑏) <s 1s ∧ 1s <s (𝐴 ·s 𝑐)) → (𝐴 ·s 𝑏) <s (𝐴 ·s 𝑐)))
92	79, 88, 91	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑏) <s (𝐴 ·s 𝑐))
93	33	ad2antrr 733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 0s <s 𝐴)
94	73, 77, 69, 93	ltmuls2d 28184	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝑏 <s 𝑐 ↔ (𝐴 ·s 𝑏) <s (𝐴 ·s 𝑐)))
95	92, 94	mpbird 259	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 <s 𝑐)
96	95	ex 414	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
97	55, 96	sylan2 600	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
98	61, 68, 97	syl2and 615	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
99	98	rexlimdvva 3198	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
100	54, 99	biimtrid 244	. . 3 ⊢ (𝜑 → ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐))
101	100	3impib 1123	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐)
102	17, 29, 38, 43, 101	sltsd 27780	1 ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055  ∃wrex 3065  {crab 3393  Vcvv 3433  ⦋csb 3832   ∪ cun 3882   ⊆ wss 3884  ∅c0 4263  {csn 4557  ⟨cop 4563  ∪ cuni 4840  ∪ ciun 4923   class class class wbr 5074   ↦ cmpt 5155   “ cima 5623   ∘ ccom 5624  Fun wfun 6482  –onto→wfo 6486  ‘cfv 6488  (class class class)co 7359  ωcom 7809  1^st c1st 7931  2^nd c2nd 7932  reccrdg 8342   No csur 27624   <s clts 27625   <<s cslts 27769   0_s c0s 27817   1_s c1s 27818   L cleft 27837   R cright 27838   +_s cadds 27971   -_s csubs 28032   ·_s cmuls 28118   /_su cdivs 28199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-dc 10364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27627  df-lts 27628  df-bday 27629  df-les 27729  df-slts 27770  df-cuts 27772  df-0s 27819  df-1s 27820  df-made 27839  df-old 27840  df-left 27842  df-right 27843  df-norec 27950  df-norec2 27961  df-adds 27972  df-negs 28033  df-subs 28034  df-muls 28119  df-divs 28200

This theorem is referenced by:  precsexlem11  28229  precsex  28230