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 7959	. . . . . . . 8 ⊢ 1st :V–onto→V
2		fofun 6751	. . . . . . . 8 ⊢ (1st :V–onto→V → Fun 1st )
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ Fun 1st
4		rdgfun 8352	. . . . . . . 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 6517	. . . . . . . 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 6536	. . . . . . 7 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
9	3, 7, 8	mp2an 693	. . . . . 6 ⊢ Fun (1st ∘ 𝐹)
10		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1st ∘ 𝐹)
11	10	funeqi 6517	. . . . . 6 ⊢ (Fun 𝐿 ↔ Fun (1st ∘ 𝐹))
12	9, 11	mpbir 231	. . . . 5 ⊢ Fun 𝐿
13		dcomex 10366	. . . . . 6 ⊢ ω ∈ V
14	13	funimaex 6584	. . . . 5 ⊢ (Fun 𝐿 → (𝐿 “ ω) ∈ V)
15	12, 14	ax-mp 5	. . . 4 ⊢ (𝐿 “ ω) ∈ V
16	15	uniex 7692	. . 3 ⊢ ∪ (𝐿 “ ω) ∈ V
17	16	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ∈ V)
18		fo2nd 7960	. . . . . . . 8 ⊢ 2nd :V–onto→V
19		fofun 6751	. . . . . . . 8 ⊢ (2nd :V–onto→V → Fun 2nd )
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ Fun 2nd
21		funco 6536	. . . . . . 7 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
22	20, 7, 21	mp2an 693	. . . . . 6 ⊢ Fun (2nd ∘ 𝐹)
23		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2nd ∘ 𝐹)
24	23	funeqi 6517	. . . . . 6 ⊢ (Fun 𝑅 ↔ Fun (2nd ∘ 𝐹))
25	22, 24	mpbir 231	. . . . 5 ⊢ Fun 𝑅
26	13	funimaex 6584	. . . . 5 ⊢ (Fun 𝑅 → (𝑅 “ ω) ∈ V)
27	25, 26	ax-mp 5	. . . 4 ⊢ (𝑅 “ ω) ∈ V
28	27	uniex 7692	. . 3 ⊢ ∪ (𝑅 “ ω) ∈ V
29	28	a1i 11	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ∈ V)
30		funiunfv 7200	. . . 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 494	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐿‘𝑖) ⊆ No )
37	36	iunssd 4994	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝐿‘𝑖) ⊆ No )
38	31, 37	eqsstrrid 3962	. 2 ⊢ (𝜑 → ∪ (𝐿 “ ω) ⊆ No )
39		funiunfv 7200	. . . 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 4994	. . 3 ⊢ (𝜑 → ∪ 𝑖 ∈ ω (𝑅‘𝑖) ⊆ No )
43	40, 42	eqsstrrid 3962	. 2 ⊢ (𝜑 → ∪ (𝑅 “ ω) ⊆ No )
44	31	eleq2i 2829	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑖 ∈ ω (𝐿‘𝑖) ↔ 𝑏 ∈ ∪ (𝐿 “ ω))
45		eliun 4938	. . . . . . 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 7200	. . . . . . . . 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 2829	. . . . . . 7 ⊢ (𝑐 ∈ ∪ 𝑗 ∈ ω (𝑅‘𝑗) ↔ 𝑐 ∈ ∪ (𝑅 “ ω))
50		eliun 4938	. . . . . . 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 629	. . . . 5 ⊢ ((𝑏 ∈ ∪ (𝐿 “ ω) ∧ 𝑐 ∈ ∪ (𝑅 “ ω)) ↔ (∃𝑖 ∈ ω 𝑏 ∈ (𝐿‘𝑖) ∧ ∃𝑗 ∈ ω 𝑐 ∈ (𝑅‘𝑗)))
53		reeanv 3210	. . . . 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 7836	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖 ∪ 𝑗) ∈ ω)
56		ssun1 4119	. . . . . . . . . 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 1453	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘𝑖) ⊆ (𝐿‘(𝑖 ∪ 𝑗)))
59	55, 58	syldan 592	. . . . . . . 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 3921	. . . . . 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 4120	. . . . . . . . . 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 1453	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
66	62, 55, 65	syl2anc 585	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘𝑗) ⊆ (𝑅‘(𝑖 ∪ 𝑗)))
67	66	sseld 3921	. . . . . . 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 727	. . . . . . . . . . . 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 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝐿‘(𝑖 ∪ 𝑗)) ⊆ No )
72	71	sselda 3922	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))) → 𝑏 ∈ No )
73	72	adantrr 718	. . . . . . . . . . . 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 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → (𝑅‘(𝑖 ∪ 𝑗)) ⊆ No )
76	75	sselda 3922	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑐 ∈ No )
77	76	adantrl 717	. . . . . . . . . . . 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 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 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 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∪ 𝑗) ∈ ω) → ∀𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗))(𝐴 ·s 𝑏) <s 1s )
82		rsp 3226	. . . . . . . . . . . . 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 3226	. . . . . . . . . . . . 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 610	. . . . . . . . . . 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		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 1452	. . . . . . . . . 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 727	. . . . . . . . . 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 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 594	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘(𝑖 ∪ 𝑗)) ∧ 𝑐 ∈ (𝑅‘(𝑖 ∪ 𝑗))) → 𝑏 <s 𝑐))
98	61, 68, 97	syl2and 609	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑏 ∈ (𝐿‘𝑖) ∧ 𝑐 ∈ (𝑅‘𝑗)) → 𝑏 <s 𝑐))
99	98	rexlimdvva 3195	. . . 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 1117	. 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 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430  ⦋csb 3838   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  ⟨cop 4574  ∪ cuni 4851  ∪ ciun 4934   class class class wbr 5086   ↦ cmpt 5167   “ cima 5631   ∘ ccom 5632  Fun wfun 6490  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7364  ωcom 7814  1^st c1st 7937  2^nd c2nd 7938  reccrdg 8345   No csur 27623   <s clts 27624   <<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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-dc 10365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-se 5582  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27626  df-lts 27627  df-bday 27628  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