precsexlem8 - Metamath Proof Explorer

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Theorem precsexlem8 28387

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 ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
precsexlem.2	⊢ 𝐿 = (1^st ∘ 𝐹)
precsexlem.3	⊢ 𝑅 = (2^nd ∘ 𝐹)
precsexlem.4	⊢ (𝜑 → 𝐴 ∈ No )
precsexlem.5	⊢ (𝜑 → 0_s <s 𝐴)
precsexlem.6	⊢ (𝜑 → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))

**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 6885	. . . . . 6 ⊢ (𝑖 = ∅ → (𝐿‘𝑖) = (𝐿‘∅))
2	1	sseq1d 3976	. . . . 5 ⊢ (𝑖 = ∅ → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘∅) ⊆ No ))
3		fveq2 6885	. . . . . 6 ⊢ (𝑖 = ∅ → (𝑅‘𝑖) = (𝑅‘∅))
4	3	sseq1d 3976	. . . . 5 ⊢ (𝑖 = ∅ → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘∅) ⊆ No ))
5	2, 4	anbi12d 643	. . . 4 ⊢ (𝑖 = ∅ → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No )))
6	5	imbi2d 343	. . 3 ⊢ (𝑖 = ∅ → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No ))))
7		fveq2 6885	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝐿‘𝑖) = (𝐿‘𝑗))
8	7	sseq1d 3976	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘𝑗) ⊆ No ))
9		fveq2 6885	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑅‘𝑖) = (𝑅‘𝑗))
10	9	sseq1d 3976	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘𝑗) ⊆ No ))
11	8, 10	anbi12d 643	. . . 4 ⊢ (𝑖 = 𝑗 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )))
12	11	imbi2d 343	. . 3 ⊢ (𝑖 = 𝑗 → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ))))
13		fveq2 6885	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝐿‘𝑖) = (𝐿‘suc 𝑗))
14	13	sseq1d 3976	. . . . 5 ⊢ (𝑖 = suc 𝑗 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘suc 𝑗) ⊆ No ))
15		fveq2 6885	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝑅‘𝑖) = (𝑅‘suc 𝑗))
16	15	sseq1d 3976	. . . . 5 ⊢ (𝑖 = suc 𝑗 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘suc 𝑗) ⊆ No ))
17	14, 16	anbi12d 643	. . . 4 ⊢ (𝑖 = suc 𝑗 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No )))
18	17	imbi2d 343	. . 3 ⊢ (𝑖 = suc 𝑗 → ((𝜑 → ((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No )) ↔ (𝜑 → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
19		fveq2 6885	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝐿‘𝑖) = (𝐿‘𝐼))
20	19	sseq1d 3976	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝐿‘𝑖) ⊆ No ↔ (𝐿‘𝐼) ⊆ No ))
21		fveq2 6885	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑅‘𝑖) = (𝑅‘𝐼))
22	21	sseq1d 3976	. . . . 5 ⊢ (𝑖 = 𝐼 → ((𝑅‘𝑖) ⊆ No ↔ (𝑅‘𝐼) ⊆ No ))
23	20, 22	anbi12d 643	. . . 4 ⊢ (𝑖 = 𝐼 → (((𝐿‘𝑖) ⊆ No ∧ (𝑅‘𝑖) ⊆ No ) ↔ ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No )))
24	23	imbi2d 343	. . 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 ↦ ⦋(1^st ‘𝑝) / 𝑙⦌⦋(2^nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ 𝑙 𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ 𝑟 𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}))⟩), ⟨{ 0_s }, ∅⟩)
26		precsexlem.2	. . . . . . 7 ⊢ 𝐿 = (1^st ∘ 𝐹)
27		precsexlem.3	. . . . . . 7 ⊢ 𝑅 = (2^nd ∘ 𝐹)
28	25, 26, 27	precsexlem1 28380	. . . . . 6 ⊢ (𝐿‘∅) = { 0_s }
29		0no 27982	. . . . . . 7 ⊢ 0_s ∈ No
30		snssi 4756	. . . . . . 7 ⊢ ( 0_s ∈ No → { 0_s } ⊆ No )
31	29, 30	ax-mp 5	. . . . . 6 ⊢ { 0_s } ⊆ No
32	28, 31	eqsstri 3991	. . . . 5 ⊢ (𝐿‘∅) ⊆ No
33	25, 26, 27	precsexlem2 28381	. . . . . 6 ⊢ (𝑅‘∅) = ∅
34		0ss 4364	. . . . . 6 ⊢ ∅ ⊆ No
35	33, 34	eqsstri 3991	. . . . 5 ⊢ (𝑅‘∅) ⊆ No
36	32, 35	pm3.2i 475	. . . 4 ⊢ ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No )
37	36	a1i 11	. . 3 ⊢ (𝜑 → ((𝐿‘∅) ⊆ No ∧ (𝑅‘∅) ⊆ No ))
38	25, 26, 27	precsexlem4 28383	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → (𝐿‘suc 𝑗) = ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
39	38	3ad2ant2 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘suc 𝑗) = ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})))
40		simp3l 1218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘𝑗) ⊆ No )
41		1no 27983	. . . . . . . . . . . . . . . 16 ⊢ 1_s ∈ No
42	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 1_s ∈ No )
43		simprl 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝑅 ∈ ( R ‘𝐴))
44	43	rightnod 28055	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝑅 ∈ No )
45		precsexlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ No )
46	45	3ad2ant1 1149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → 𝐴 ∈ No )
47	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
48	44, 47	subscld 28236	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝑥_𝑅 -_s 𝐴) ∈ No )
49		simpl3l 1245	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝐿‘𝑗) ⊆ No )
50		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑦_𝐿 ∈ (𝐿‘𝑗))
51	49, 50	sseldd 3946	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑦_𝐿 ∈ No )
52	48, 51	mulscld 28308	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿) ∈ No )
53	42, 52	addscld 28153	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) ∈ No )
54	29	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 0_s ∈ No )
55		precsexlem.5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0_s <s 𝐴)
56	55	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → 0_s <s 𝐴)
57	56	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 0_s <s 𝐴)
58		rightgt 28027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑅 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥_𝑅)
59	43, 58	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝐴 <s 𝑥_𝑅)
60	54, 47, 44, 57, 59	ltstrd 27907	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 0_s <s 𝑥_𝑅)
61	60	gt0ne0sd 27992	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝑅 ≠ 0_s )
62		breq2 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑂 = 𝑥_𝑅 → ( 0_s <s 𝑥_𝑂 ↔ 0_s <s 𝑥_𝑅))
63		oveq1 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥_𝑂 = 𝑥_𝑅 → (𝑥_𝑂 ·_s 𝑦) = (𝑥_𝑅 ·_s 𝑦))
64	63	eqeq1d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥_𝑂 = 𝑥_𝑅 → ((𝑥_𝑂 ·_s 𝑦) = 1_s ↔ (𝑥_𝑅 ·_s 𝑦) = 1_s ))
65	64	rexbidv 3196	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑂 = 𝑥_𝑅 → (∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s ))
66	62, 65	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑥_𝑂 = 𝑥_𝑅 → (( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ) ↔ ( 0_s <s 𝑥_𝑅 → ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s )))
67		precsexlem.6	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
68	67	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
69	68	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
70		elun2 4144	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑅 ∈ ( R ‘𝐴) → 𝑥_𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
71	43, 70	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
72	66, 69, 71	rspcdva 3590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ( 0_s <s 𝑥_𝑅 → ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s ))
73	60, 72	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s )
74	53, 44, 61, 73	divsclwd 28369	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) ∈ No )
75		eleq1 2858	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) → (𝑎 ∈ No ↔ (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) ∈ No ))
76	74, 75	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) → 𝑎 ∈ No ))
77	76	rexlimdvva 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅) → 𝑎 ∈ No ))
78	77	abssdv 4029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ⊆ No )
79	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 1_s ∈ No )
80		ssrab2 4042	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ⊆ ( L ‘𝐴)
81		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})
82	80, 81	sselid 3943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝐿 ∈ ( L ‘𝐴))
83	82	leftnod 28053	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝐿 ∈ No )
84	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
85	83, 84	subscld 28236	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑥_𝐿 -_s 𝐴) ∈ No )
86		simpl3r 1246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑅‘𝑗) ⊆ No )
87		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑦_𝑅 ∈ (𝑅‘𝑗))
88	86, 87	sseldd 3946	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑦_𝑅 ∈ No )
89	85, 88	mulscld 28308	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅) ∈ No )
90	79, 89	addscld 28153	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) ∈ No )
91		breq2 5118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑥_𝐿 → ( 0_s <s 𝑥 ↔ 0_s <s 𝑥_𝐿))
92	91	elrab 3658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ↔ (𝑥_𝐿 ∈ ( L ‘𝐴) ∧ 0_s <s 𝑥_𝐿))
93	92	simprbi 502	. . . . . . . . . . . . . . . 16 ⊢ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} → 0_s <s 𝑥_𝐿)
94	81, 93	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 0_s <s 𝑥_𝐿)
95	94	gt0ne0sd 27992	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝐿 ≠ 0_s )
96		breq2 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑂 = 𝑥_𝐿 → ( 0_s <s 𝑥_𝑂 ↔ 0_s <s 𝑥_𝐿))
97		oveq1 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥_𝑂 = 𝑥_𝐿 → (𝑥_𝑂 ·_s 𝑦) = (𝑥_𝐿 ·_s 𝑦))
98	97	eqeq1d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥_𝑂 = 𝑥_𝐿 → ((𝑥_𝑂 ·_s 𝑦) = 1_s ↔ (𝑥_𝐿 ·_s 𝑦) = 1_s ))
99	98	rexbidv 3196	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝑂 = 𝑥_𝐿 → (∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ↔ ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s ))
100	96, 99	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑥_𝑂 = 𝑥_𝐿 → (( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ) ↔ ( 0_s <s 𝑥_𝐿 → ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s )))
101	68	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
102		elun1 4143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥_𝐿 ∈ ( L ‘𝐴) → 𝑥_𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
103	82, 102	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
104	100, 101, 103	rspcdva 3590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ( 0_s <s 𝑥_𝐿 → ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s ))
105	94, 104	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s )
106	90, 83, 95, 105	divsclwd 28369	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) ∈ No )
107		eleq1 2858	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) → (𝑎 ∈ No ↔ (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) ∈ No ))
108	106, 107	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) → 𝑎 ∈ No ))
109	108	rexlimdvva 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿) → 𝑎 ∈ No ))
110	109	abssdv 4029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)} ⊆ No )
111	78, 110	unssd 4153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)}) ⊆ No )
112	40, 111	unssd 4153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝐿‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝑅)} ∪ {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝐿)})) ⊆ No )
113	39, 112	eqsstrd 3979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝐿‘suc 𝑗) ⊆ No )
114	25, 26, 27	precsexlem5 28384	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → (𝑅‘suc 𝑗) = ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
115	114	3ad2ant2 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝑅‘suc 𝑗) = ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})))
116		simp3r 1219	. . . . . . . . 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 )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 1_s ∈ No )
118		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥})
119	80, 118	sselid 3943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝐿 ∈ ( L ‘𝐴))
120	119	leftnod 28053	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝐿 ∈ No )
121	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
122	120, 121	subscld 28236	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝑥_𝐿 -_s 𝐴) ∈ No )
123		simpl3l 1245	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝐿‘𝑗) ⊆ No )
124		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑦_𝐿 ∈ (𝐿‘𝑗))
125	123, 124	sseldd 3946	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑦_𝐿 ∈ No )
126	122, 125	mulscld 28308	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿) ∈ No )
127	117, 126	addscld 28153	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) ∈ No )
128	118, 93	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 0_s <s 𝑥_𝐿)
129	128	gt0ne0sd 27992	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝐿 ≠ 0_s )
130	68	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
131	119, 102	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → 𝑥_𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
132	100, 130, 131	rspcdva 3590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ( 0_s <s 𝑥_𝐿 → ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s ))
133	128, 132	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥_𝐿 ·_s 𝑦) = 1_s )
134	127, 120, 129, 133	divsclwd 28369	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) ∈ No )
135		eleq1 2858	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) → (𝑎 ∈ No ↔ (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) ∈ No ))
136	134, 135	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥} ∧ 𝑦_𝐿 ∈ (𝐿‘𝑗))) → (𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) → 𝑎 ∈ No ))
137	136	rexlimdvva 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿) → 𝑎 ∈ No ))
138	137	abssdv 4029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ⊆ No )
139	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 1_s ∈ No )
140		simprl 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝑅 ∈ ( R ‘𝐴))
141	140	rightnod 28055	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝑅 ∈ No )
142	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
143	141, 142	subscld 28236	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑥_𝑅 -_s 𝐴) ∈ No )
144		simpl3r 1246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑅‘𝑗) ⊆ No )
145		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑦_𝑅 ∈ (𝑅‘𝑗))
146	144, 145	sseldd 3946	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑦_𝑅 ∈ No )
147	143, 146	mulscld 28308	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅) ∈ No )
148	139, 147	addscld 28153	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) ∈ No )
149	29	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 0_s ∈ No )
150	56	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 0_s <s 𝐴)
151	140, 58	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝐴 <s 𝑥_𝑅)
152	149, 142, 141, 150, 151	ltstrd 27907	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 0_s <s 𝑥_𝑅)
153	152	gt0ne0sd 27992	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝑅 ≠ 0_s )
154	68	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0_s <s 𝑥_𝑂 → ∃𝑦 ∈ No (𝑥_𝑂 ·_s 𝑦) = 1_s ))
155	140, 70	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → 𝑥_𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
156	66, 154, 155	rspcdva 3590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ( 0_s <s 𝑥_𝑅 → ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s ))
157	152, 156	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥_𝑅 ·_s 𝑦) = 1_s )
158	148, 141, 153, 157	divsclwd 28369	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) ∈ No )
159		eleq1 2858	. . . . . . . . . . . . 13 ⊢ (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) → (𝑎 ∈ No ↔ (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) ∈ No ))
160	158, 159	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) ∧ (𝑥_𝑅 ∈ ( R ‘𝐴) ∧ 𝑦_𝑅 ∈ (𝑅‘𝑗))) → (𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) → 𝑎 ∈ No ))
161	160	rexlimdvva 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅) → 𝑎 ∈ No ))
162	161	abssdv 4029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)} ⊆ No )
163	138, 162	unssd 4153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)}) ⊆ No )
164	116, 163	unssd 4153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝑅‘𝑗) ∪ ({𝑎 ∣ ∃𝑥_𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0_s <s 𝑥}∃𝑦_𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝐿 -_s 𝐴) ·_s 𝑦_𝐿)) /_su 𝑥_𝐿)} ∪ {𝑎 ∣ ∃𝑥_𝑅 ∈ ( R ‘𝐴)∃𝑦_𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1_s +_s ((𝑥_𝑅 -_s 𝐴) ·_s 𝑦_𝑅)) /_su 𝑥_𝑅)})) ⊆ No )
165	115, 164	eqsstrd 3979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → (𝑅‘suc 𝑗) ⊆ No )
166	113, 165	jca 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No )) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))
167	166	3exp 1135	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ω → (((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
168	167	com12 33	. . . 4 ⊢ (𝑗 ∈ ω → (𝜑 → (((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ) → ((𝐿‘suc 𝑗) ⊆ No ∧ (𝑅‘suc 𝑗) ⊆ No ))))
169	168	a2d 30	. . 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 7896	. 2 ⊢ (𝐼 ∈ ω → (𝜑 → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No )))
171	170	impcom 412	1 ⊢ ((𝜑 ∧ 𝐼 ∈ ω) → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2150 {cab 2748 ∀wral 3086 ∃wrex 3096 {crab 3423 Vcvv 3462 ⦋csb 3861 ∪ cun 3911 ⊆ wss 3913 ∅c0 4294 {csn 4594 ⟨cop 4600 class class class wbr 5114 ↦ cmpt 5197 ∘ ccom 5669 suc csuc 6366 ‘cfv 6540 (class class class)co 7414 ωcom 7865 1^st c1st 7987 2^nd c2nd 7988 reccrdg 8399 No csur 27784 <s clts 27785 0_s c0s 27978 1_s c1s 27979 L cleft 27998 R cright 27999 +_s cadds 28132 -_s csubs 28193 ·_s cmuls 28279 /_su cdivs 28360

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-nadd 8655 df-no 27787 df-lts 27788 df-bday 27789 df-les 27889 df-slts 27931 df-cuts 27933 df-0s 27980 df-1s 27981 df-made 28000 df-old 28001 df-left 28003 df-right 28004 df-norec 28111 df-norec2 28122 df-adds 28133 df-negs 28194 df-subs 28195 df-muls 28280 df-divs 28361

This theorem is referenced by: precsexlem9 28388 precsexlem10 28389

W3C validator