Theorem precsexlem10 28125

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 7944	. . . . . . . 8 ⊢ 1st :V–onto→V
2		fofun 6737	. . . . . . . 8 ⊢ (1st :V–onto→V → Fun 1st )
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ Fun 1st
4		rdgfun 8338	. . . . . . . 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 6503	. . . . . . . 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 231	. . . . . . 7 ⊢ Fun 𝐹
8		funco 6522	. . . . . . 7 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
9	3, 7, 8	mp2an 692	. . . . . 6 ⊢ Fun (1st ∘ 𝐹)
10		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1st ∘ 𝐹)
11	10	funeqi 6503	. . . . . 6 ⊢ (Fun 𝐿 ↔ Fun (1st ∘ 𝐹))
12	9, 11	mpbir 231	. . . . 5 ⊢ Fun 𝐿
13		dcomex 10341	. . . . . 6 ⊢ ω ∈ V
14	13	funimaex 6570	. . . . 5 ⊢ (Fun 𝐿 → (𝐿 “ ω) ∈ V)
15	12, 14	ax-mp 5	. . . 4 ⊢ (𝐿 “ ω) ∈ V
16	15	uniex 7677	. . 3 ⊢ ∪ (𝐿 “ ω) ∈ V
17	16	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ∈ V)
18		fo2nd 7945	. . . . . . . 8 ⊢ 2nd :V–onto→V
19		fofun 6737	. . . . . . . 8 ⊢ (2nd :V–onto→V → Fun 2nd )
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ Fun 2nd
21		funco 6522	. . . . . . 7 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
22	20, 7, 21	mp2an 692	. . . . . 6 ⊢ Fun (2nd ∘ 𝐹)
23		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2nd ∘ 𝐹)
24	23	funeqi 6503	. . . . . 6 ⊢ (Fun 𝑅 ↔ Fun (2nd ∘ 𝐹))
25	22, 24	mpbir 231	. . . . 5 ⊢ Fun 𝑅
26	13	funimaex 6570	. . . . 5 ⊢ (Fun 𝑅 → (𝑅 “ ω) ∈ V)
27	25, 26	ax-mp 5	. . . 4 ⊢ (𝑅 “ ω) ∈ V
28	27	uniex 7677	. . 3 ⊢ ∪ (𝑅 “ ω) ∈ V
29	28	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ∈ V)
30		funiunfv 7184	. . . 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 28123	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ))
36	35	simpld 494	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐿‘𝑖) ⊆ No )
37	36	iunssd 4999	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) ⊆ No )
38	31, 37	eqsstrrid 3975	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ⊆ No )
39		funiunfv 7184	. . . 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 495	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑅‘𝑖) ⊆ No )
42	41	iunssd 4999	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) ⊆ No )
43	40, 42	eqsstrrid 3975	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ⊆ No )
44	31	eleq2i 2820	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ 𝑏 ∈ ∪ (𝐿 “ ω))
45		eliun 4945	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
46	44, 45	bitr3i 277	. . . . . 6 ⊢ (𝑏 ∈ ∪ (𝐿 “ ω) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
47		funiunfv 7184	. . . . . . . . 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 2820	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ 𝑐 ∈ ∪ (𝑅 “ ω))
50		eliun 4945	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
51	49, 50	bitr3i 277	. . . . . 6 ⊢ (𝑐 ∈ ∪ (𝑅 “ ω) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
52	46, 51	anbi12i 628	. . . . 5 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ (∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖) ∧ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗)))
53		reeanv 3201	. . . . 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 278	. . . 4 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)))
55		omun 7821	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖 ∪ 𝑗) ∈ ω)
56		ssun1 4129	. . . . . . . . . 10 ⊢ 𝑖 ⊆ (𝑖 ∪ 𝑗)
57	5, 10, 23	precsexlem6 28121	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑖 ⊆ (𝑖 ∪ 𝑗)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
58	56, 57	mp3an3 1452	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
59	55, 58	syldan 591	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
60	59	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
61	60	sseld 3934	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑏 ∈ (𝐿‘𝑖) → 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))))
62		simpr 484	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
63		ssun2 4130	. . . . . . . . . 10 ⊢ 𝑗 ⊆ (𝑖 ∪ 𝑗)
64	5, 10, 23	precsexlem7 28122	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑗 ⊆ (𝑖 ∪ 𝑗)) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
65	63, 64	mp3an3 1452	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
66	62, 55, 65	syl2anc 584	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
67	66	sseld 3934	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
68	67	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
69	32	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝐴 ∈ No )
70	5, 10, 23, 32, 33, 34	precsexlem8 28123	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝐿‘(𝑖 ∪ 𝑗)) ⊆ No ∧ (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No ))
71	70	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘(𝑖 ∪ 𝑗)) ⊆ No )
72	71	sselda 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))) → 𝑏 ∈ No )
73	72	adantrr 717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 ∈ No )
74	69, 73	mulscld 28045	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑏) ∈ No )
75	70	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No )
76	75	sselda 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑐 ∈ No )
77	76	adantrl 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑐 ∈ No )
78	69, 77	mulscld 28045	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑐) ∈ No )
79	74, 78	jca 511	. . . . . . . . . 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 28124	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s )
82		rsp 3217	. . . . . . . . . . . . 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 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐))
85		rsp 3217	. . . . . . . . . . . . 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 609	. . . . . . . . . . 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 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → ((𝐴 ·s 𝑏) <s 1s ∧ 1s <s (𝐴 ·s 𝑐)))
89		1sno 27742	. . . . . . . . . . 11 ⊢ 1s ∈ No
90		slttr 27657	. . . . . . . . . . 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 1451	. . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 0s <s 𝐴)
94	73, 77, 69, 93	sltmul2d 28082	. . . . . . . . 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 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 <s 𝑐)
96	95	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
97	55, 96	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
98	61, 68, 97	syl2and 608	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
99	98	rexlimdvva 3186	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
100	54, 99	biimtrid 242	. . 3 ⊢ (𝜑 → ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐))
101	100	3impib 1116	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐)
102	17, 29, 38, 43, 101	ssltd 27702	1 ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436  ⦋csb 3851   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577  ⟨cop 4583  ∪ cuni 4858  ∪ ciun 4941   class class class wbr 5092   ↦ cmpt 5173   “ cima 5622   ∘ ccom 5623  Fun wfun 6476  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349  ωcom 7799  1^st c1st 7922  2^nd c2nd 7923  reccrdg 8331   No csur 27549   <s cslt 27550   <<s csslt 27691   0_s c0s 27737   1_s c1s 27738   L cleft 27757   R cright 27758   +_s cadds 27873   -_s csubs 27933   ·_s cmuls 28016   /_su cdivs 28097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-dc 10340

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017  df-divs 28098

This theorem is referenced by:  precsexlem11  28126  precsex  28127