Theorem precsexlem10 28420

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 8004	. . . . . . . 8 ⊢ 1st :V–onto→V
2		fofun 6793	. . . . . . . 8 ⊢ (1st :V–onto→V → Fun 1st )
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ Fun 1st
4		rdgfun 8401	. . . . . . . 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 6557	. . . . . . . 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 234	. . . . . . 7 ⊢ Fun 𝐹
8		funco 6576	. . . . . . 7 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
9	3, 7, 8	mp2an 704	. . . . . 6 ⊢ Fun (1st ∘ 𝐹)
10		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1st ∘ 𝐹)
11	10	funeqi 6557	. . . . . 6 ⊢ (Fun 𝐿 ↔ Fun (1st ∘ 𝐹))
12	9, 11	mpbir 234	. . . . 5 ⊢ Fun 𝐿
13		dcomex 10437	. . . . . 6 ⊢ ω ∈ V
14	13	funimaex 6623	. . . . 5 ⊢ (Fun 𝐿 → (𝐿 “ ω) ∈ V)
15	12, 14	ax-mp 5	. . . 4 ⊢ (𝐿 “ ω) ∈ V
16	15	uniex 7741	. . 3 ⊢ ∪ (𝐿 “ ω) ∈ V
17	16	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ∈ V)
18		fo2nd 8005	. . . . . . . 8 ⊢ 2nd :V–onto→V
19		fofun 6793	. . . . . . . 8 ⊢ (2nd :V–onto→V → Fun 2nd )
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ Fun 2nd
21		funco 6576	. . . . . . 7 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
22	20, 7, 21	mp2an 704	. . . . . 6 ⊢ Fun (2nd ∘ 𝐹)
23		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2nd ∘ 𝐹)
24	23	funeqi 6557	. . . . . 6 ⊢ (Fun 𝑅 ↔ Fun (2nd ∘ 𝐹))
25	22, 24	mpbir 234	. . . . 5 ⊢ Fun 𝑅
26	13	funimaex 6623	. . . . 5 ⊢ (Fun 𝑅 → (𝑅 “ ω) ∈ V)
27	25, 26	ax-mp 5	. . . 4 ⊢ (𝑅 “ ω) ∈ V
28	27	uniex 7741	. . 3 ⊢ ∪ (𝑅 “ ω) ∈ V
29	28	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ∈ V)
30		funiunfv 7246	. . . 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 28418	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ))
36	35	simpld 499	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐿‘𝑖) ⊆ No )
37	36	iunssd 5014	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) ⊆ No )
38	31, 37	eqsstrrid 3975	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ⊆ No )
39		funiunfv 7246	. . . 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 500	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑅‘𝑖) ⊆ No )
42	41	iunssd 5014	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) ⊆ No )
43	40, 42	eqsstrrid 3975	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ⊆ No )
44	31	eleq2i 2854	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ 𝑏 ∈ ∪ (𝐿 “ ω))
45		eliun 4959	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
46	44, 45	bitr3i 280	. . . . . 6 ⊢ (𝑏 ∈ ∪ (𝐿 “ ω) ↔ ∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖))
47		funiunfv 7246	. . . . . . . . 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 2854	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ 𝑐 ∈ ∪ (𝑅 “ ω))
50		eliun 4959	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
51	49, 50	bitr3i 280	. . . . . 6 ⊢ (𝑐 ∈ ∪ (𝑅 “ ω) ↔ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗))
52	46, 51	anbi12i 639	. . . . 5 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ (∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖) ∧ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗)))
53		reeanv 3236	. . . . 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 281	. . . 4 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)))
55		omun 7882	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖 ∪ 𝑗) ∈ ω)
56		ssun1 4130	. . . . . . . . . 10 ⊢ 𝑖 ⊆ (𝑖 ∪ 𝑗)
57	5, 10, 23	precsexlem6 28416	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑖 ⊆ (𝑖 ∪ 𝑗)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
58	56, 57	mp3an3 1478	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
59	55, 58	syldan 602	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
60	59	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
61	60	sseld 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑏 ∈ (𝐿‘𝑖) → 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))))
62		simpr 489	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
63		ssun2 4131	. . . . . . . . . 10 ⊢ 𝑗 ⊆ (𝑖 ∪ 𝑗)
64	5, 10, 23	precsexlem7 28417	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω ∧ 𝑗 ⊆ (𝑖 ∪ 𝑗)) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
65	63, 64	mp3an3 1478	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
66	62, 55, 65	syl2anc 595	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
67	66	sseld 3935	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
68	67	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑐 ∈ (𝑅‘𝑗) → 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))))
69	32	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝐴 ∈ No )
70	5, 10, 23, 32, 33, 34	precsexlem8 28418	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝐿‘(𝑖 ∪ 𝑗)) ⊆ No ∧ (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No ))
71	70	simpld 499	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘(𝑖 ∪ 𝑗)) ⊆ No )
72	71	sselda 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))) → 𝑏 ∈ No )
73	72	adantrr 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 ∈ No )
74	69, 73	mulscld 28339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑏) ∈ No )
75	70	simprd 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No )
76	75	sselda 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑐 ∈ No )
77	76	adantrl 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑐 ∈ No )
78	69, 77	mulscld 28339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑐) ∈ No )
79	74, 78	jca 520	. . . . . . . . . 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 28419	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s )
82		rsp 3252	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s → (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) → (𝐴 ·s 𝑏) <s 1s ))
83	81, 82	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) → (𝐴 ·s 𝑏) <s 1s ))
84	80	simprd 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐))
85		rsp 3252	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) 1s <s (𝐴 ·s 𝑐) → (𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) → 1s <s (𝐴 ·s 𝑐)))
86	84, 85	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)) → 1s <s (𝐴 ·s 𝑐)))
87	83, 86	anim12d 620	. . . . . . . . . . 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 411	. . . . . . . . . 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 28014	. . . . . . . . . . 11 ⊢ 1s ∈ No
90		ltstr 27922	. . . . . . . . . . 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 1477	. . . . . . . . . 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 66	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → (𝐴 ·s 𝑏) <s (𝐴 ·s 𝑐))
93	33	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 0s <s 𝐴)
94	73, 77, 69, 93	ltmuls2d 28376	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ (𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗)))) → 𝑏 <s 𝑐)
96	95	ex 417	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
97	55, 96	sylan2 604	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
98	61, 68, 97	syl2and 619	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
99	98	rexlimdvva 3221	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
100	54, 99	biimtrid 245	. . 3 ⊢ (𝜑 → ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐))
101	100	3impib 1133	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) → 𝑏 <s 𝑐)
102	17, 29, 38, 43, 101	sltsd 27972	1 ⊢ (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  {crab 3415  Vcvv 3454  ⦋csb 3852   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4588  ⟨cop 4594  ∪ cuni 4871  ∪ ciun 4955   class class class wbr 5108   ↦ cmpt 5191   “ cima 5663   ∘ ccom 5664  Fun wfun 6530  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7412  ωcom 7860  1^st c1st 7982  2^nd c2nd 7983  reccrdg 8394   No csur 27815   <s clts 27816   <<s cslts 27961   0_s c0s 28009   1_s c1s 28010   L cleft 28029   R cright 28030   +_s cadds 28163   -_s csubs 28224   ·_s cmuls 28310   /_su cdivs 28391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-dc 10436

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-divs 28392

This theorem is used by:  precsexlem11  28421  precsex  28422