Theorem precsexlem8 28309

Description: Lemma for surreal reciprocal. Show that the left and right functions give sets of surreals. (Contributed by Scott Fenton, 13-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
precsexlem8	⊢ ((𝜑 ∧ 𝐼 ∈ ω) → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No ))

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑟,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)   𝐼(𝑥,𝑦,𝑟,𝑝,𝑎,𝑙,𝑥_𝑂,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅)   𝐿(𝑥,𝑦,𝑟,𝑝,𝑥_𝑂)

Proof of Theorem precsexlem8

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

Step	Hyp	Ref	Expression
1		fveq2 6869	. . . . . 6 ⊢ (𝑖 = ∅ → (𝐿‘𝑖) = (𝐿‘∅))
2	1	sseq1d 3969	. . . . 5 ⊢ (𝑖 = ∅ → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘∅) ⊆ No ))
3		fveq2 6869	. . . . . 6 ⊢ (𝑖 = ∅ → (𝑅‘𝑖) = (𝑅‘∅))
4	3	sseq1d 3969	. . . . 5 ⊢ (𝑖 = ∅ → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘∅) ⊆ No ))
5	2, 4	anbi12d 641	. . . 4 ⊢ (𝑖 = ∅ → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No )))
6	5	imbi2d 342	. . 3 ⊢ (𝑖 = ∅ → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No ))))
7		fveq2 6869	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝐿‘𝑖) = (𝐿‘𝑗))
8	7	sseq1d 3969	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘𝑗) ⊆ No ))
9		fveq2 6869	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑅‘𝑖) = (𝑅‘𝑗))
10	9	sseq1d 3969	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘𝑗) ⊆ No ))
11	8, 10	anbi12d 641	. . . 4 ⊢ (𝑖 = 𝑗 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )))
12	11	imbi2d 342	. . 3 ⊢ (𝑖 = 𝑗 → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ))))
13		fveq2 6869	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝐿‘𝑖) = (𝐿‘suc 𝑗))
14	13	sseq1d 3969	. . . . 5 ⊢ (𝑖 = suc 𝑗 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘suc 𝑗) ⊆ No ))
15		fveq2 6869	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝑅‘𝑖) = (𝑅‘suc 𝑗))
16	15	sseq1d 3969	. . . . 5 ⊢ (𝑖 = suc 𝑗 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘suc 𝑗) ⊆ No ))
17	14, 16	anbi12d 641	. . . 4 ⊢ (𝑖 = suc 𝑗 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No )))
18	17	imbi2d 342	. . 3 ⊢ (𝑖 = suc 𝑗 → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
19		fveq2 6869	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝐿‘𝑖) = (𝐿‘𝐼))
20	19	sseq1d 3969	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘𝐼) ⊆ No ))
21		fveq2 6869	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑅‘𝑖) = (𝑅‘𝐼))
22	21	sseq1d 3969	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘𝐼) ⊆ No ))
23	20, 22	anbi12d 641	. . . 4 ⊢ (𝑖 = 𝐼 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No )))
24	23	imbi2d 342	. . 3 ⊢ (𝑖 = 𝐼 → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No ))))
25		precsexlem.1	. . . . . . 7 ⊢ 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 }, ∅⟩)
26		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1st ∘ 𝐹)
27		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2nd ∘ 𝐹)
28	25, 26, 27	precsexlem1 28302	. . . . . 6 ⊢ (𝐿‘∅) = { 0s }
29		0no 27904	. . . . . . 7 ⊢ 0s ∈ No
30		snssi 4746	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ⊆ No )
31	29, 30	ax-mp 5	. . . . . 6 ⊢ { 0s } ⊆ No
32	28, 31	eqsstri 3984	. . . . 5 ⊢ (𝐿‘∅) ⊆ No
33	25, 26, 27	precsexlem2 28303	. . . . . 6 ⊢ (𝑅‘∅) = ∅
34		0ss 4356	. . . . . 6 ⊢ ∅ ⊆ No
35	33, 34	eqsstri 3984	. . . . 5 ⊢ (𝑅‘∅) ⊆ No
36	32, 35	pm3.2i 474	. . . 4 ⊢ ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No )
37	36	a1i 11	. . 3 ⊢ (𝜑 → ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No ))
38	25, 26, 27	precsexlem4 28305	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → (𝐿‘suc 𝑗) = ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
39	38	3ad2ant2 1148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘suc 𝑗) = ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
40		simp3l 1216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘𝑗) ⊆ No )
41		1no 27905	. . . . . . . . . . . . . . . 16 ⊢ 1s ∈ No
42	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s ∈ No )
43		simprl 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ∈ ( R ‘𝐴))
44	43	rightnod 27977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ∈ No )
45		precsexlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ No )
46	45	3ad2ant1 1147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → 𝐴 ∈ No )
47	46	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
48	44, 47	subscld 28158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑥𝑅 -s 𝐴) ∈ No )
49		simpl3l 1243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐿‘𝑗) ⊆ No )
50		simprr 782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ (𝐿‘𝑗))
51	49, 50	sseldd 3939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ No )
52	48, 51	mulscld 28230	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿) ∈ No )
53	42, 52	addscld 28075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
54	29	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s ∈ No )
55		precsexlem.5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0s <s 𝐴)
56	55	3ad2ant1 1147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → 0s <s 𝐴)
57	56	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s 𝐴)
58		rightgt 27949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑅)
59	43, 58	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝐴 <s 𝑥𝑅)
60	54, 47, 44, 57, 59	ltstrd 27829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s 𝑥𝑅)
61	60	gt0ne0sd 27914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ≠ 0s )
62		breq2 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝑅 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝑅))
63		oveq1 7405	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑅 ·s 𝑦))
64	63	eqeq1d 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝑅 ·s 𝑦) = 1s ))
65	64	rexbidv 3188	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝑅 → (∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s ))
66	62, 65	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 = 𝑥𝑅 → (( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝑅 → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )))
67		precsexlem.6	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
68	67	3ad2ant1 1147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
69	68	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
70		elun2 4137	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
71	43, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
72	66, 69, 71	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 0s <s 𝑥𝑅 → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s ))
73	60, 72	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )
74	53, 44, 61, 73	divsclwd 28291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∈ No )
75		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝑎 ∈ No ↔ (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∈ No ))
76	74, 75	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → 𝑎 ∈ No ))
77	76	rexlimdvva 3221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → 𝑎 ∈ No ))
78	77	abssdv 4022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ⊆ No )
79	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s ∈ No )
80		ssrab2 4035	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ⊆ ( L ‘𝐴)
81		simprl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥})
82	80, 81	sselid 3936	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ∈ ( L ‘𝐴))
83	82	leftnod 27975	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ∈ No )
84	46	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
85	83, 84	subscld 28158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
86		simpl3r 1244	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑅‘𝑗) ⊆ No )
87		simprr 782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ (𝑅‘𝑗))
88	86, 87	sseldd 3939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ No )
89	85, 88	mulscld 28230	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅) ∈ No )
90	79, 89	addscld 28075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
91		breq2 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑥𝐿 → ( 0s <s 𝑥 ↔ 0s <s 𝑥𝐿))
92	91	elrab 3652	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ↔ (𝑥𝐿 ∈ ( L ‘𝐴) ∧ 0s <s 𝑥𝐿))
93	92	simprbi 501	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} → 0s <s 𝑥𝐿)
94	81, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s 𝑥𝐿)
95	94	gt0ne0sd 27914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ≠ 0s )
96		breq2 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝐿 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝐿))
97		oveq1 7405	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 ·s 𝑦) = (𝑥𝐿 ·s 𝑦))
98	97	eqeq1d 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝐿 ·s 𝑦) = 1s ))
99	98	rexbidv 3188	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = 𝑥𝐿 → (∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s ))
100	96, 99	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 = 𝑥𝐿 → (( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝐿 → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )))
101	68	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
102		elun1 4136	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
103	82, 102	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
104	100, 101, 103	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 0s <s 𝑥𝐿 → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s ))
105	94, 104	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )
106	90, 83, 95, 105	divsclwd 28291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ∈ No )
107		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝑎 ∈ No ↔ (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ∈ No ))
108	106, 107	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → 𝑎 ∈ No ))
109	108	rexlimdvva 3221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → 𝑎 ∈ No ))
110	109	abssdv 4022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)} ⊆ No )
111	78, 110	unssd 4146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}) ⊆ No )
112	40, 111	unssd 4146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})) ⊆ No )
113	39, 112	eqsstrd 3972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘suc 𝑗) ⊆ No )
114	25, 26, 27	precsexlem5 28306	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → (𝑅‘suc 𝑗) = ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
115	114	3ad2ant2 1148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝑅‘suc 𝑗) = ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
116		simp3r 1217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝑅‘𝑗) ⊆ No )
117	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s ∈ No )
118		simprl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥})
119	80, 118	sselid 3936	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ∈ ( L ‘𝐴))
120	119	leftnod 27975	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ∈ No )
121	46	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
122	120, 121	subscld 28158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
123		simpl3l 1243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐿‘𝑗) ⊆ No )
124		simprr 782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ (𝐿‘𝑗))
125	123, 124	sseldd 3939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ No )
126	122, 125	mulscld 28230	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿) ∈ No )
127	117, 126	addscld 28075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
128	118, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s 𝑥𝐿)
129	128	gt0ne0sd 27914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ≠ 0s )
130	68	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
131	119, 102	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
132	100, 130, 131	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 0s <s 𝑥𝐿 → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s ))
133	128, 132	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )
134	127, 120, 129, 133	divsclwd 28291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∈ No )
135		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → (𝑎 ∈ No ↔ (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∈ No ))
136	134, 135	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 𝑎 ∈ No ))
137	136	rexlimdvva 3221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 𝑎 ∈ No ))
138	137	abssdv 4022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ⊆ No )
139	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s ∈ No )
140		simprl 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ∈ ( R ‘𝐴))
141	140	rightnod 27977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ∈ No )
142	46	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
143	141, 142	subscld 28158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝑅 -s 𝐴) ∈ No )
144		simpl3r 1244	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑅‘𝑗) ⊆ No )
145		simprr 782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ (𝑅‘𝑗))
146	144, 145	sseldd 3939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ No )
147	143, 146	mulscld 28230	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅) ∈ No )
148	139, 147	addscld 28075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
149	29	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s ∈ No )
150	56	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s 𝐴)
151	140, 58	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝐴 <s 𝑥𝑅)
152	149, 142, 141, 150, 151	ltstrd 27829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s 𝑥𝑅)
153	152	gt0ne0sd 27914	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ≠ 0s )
154	68	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
155	140, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
156	66, 154, 155	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 0s <s 𝑥𝑅 → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s ))
157	152, 156	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )
158	148, 141, 153, 157	divsclwd 28291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ∈ No )
159		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → (𝑎 ∈ No ↔ (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ∈ No ))
160	158, 159	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 𝑎 ∈ No ))
161	160	rexlimdvva 3221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 𝑎 ∈ No ))
162	161	abssdv 4022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)} ⊆ No )
163	138, 162	unssd 4146	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}) ⊆ No )
164	116, 163	unssd 4146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})) ⊆ No )
165	115, 164	eqsstrd 3972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝑅‘suc 𝑗) ⊆ No )
166	113, 165	jca 519	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))
167	166	3exp 1133	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ω → (((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
168	167	com12 32	. . . 4 ⊢ (𝑗 ∈ ω → (𝜑 → (((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
169	168	a2d 29	. . 3 ⊢ (𝑗 ∈ ω → ((𝜑 → ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝜑 → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
170	6, 12, 18, 24, 37, 169	finds 7879	. 2 ⊢ (𝐼 ∈ ω → (𝜑 → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No )))
171	170	impcom 411	1 ⊢ ((𝜑 ∧ 𝐼 ∈ ω) → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No ))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088  {crab 3416  Vcvv 3456  ⦋csb 3854   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183   ∘ ccom 5653  suc csuc 6350  ‘cfv 6523  (class class class)co 7398  ωcom 7848  1^st c1st 7970  2^nd c2nd 7971  reccrdg 8382   No csur 27706   <s clts 27707   0_s c0s 27900   1_s c1s 27901   L cleft 27920   R cright 27921   +_s cadds 28054   -_s csubs 28115   ·_s cmuls 28201   /_su cdivs 28282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-nadd 8638  df-no 27709  df-lts 27710  df-bday 27711  df-les 27811  df-slts 27853  df-cuts 27855  df-0s 27902  df-1s 27903  df-made 27922  df-old 27923  df-left 27925  df-right 27926  df-norec 28033  df-norec2 28044  df-adds 28055  df-negs 28116  df-subs 28117  df-muls 28202  df-divs 28283

This theorem is referenced by:  precsexlem9  28310  precsexlem10  28311