Theorem precsexlem9 28239

Description: Lemma for surreal reciprocal. Show that the product of 𝐴 and a left element is less than one and the product of 𝐴 and a right element is greater than one. (Contributed by Scott Fenton, 14-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
precsexlem9	⊢ ((𝜑 ∧ 𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))

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   𝐴,𝑏,𝑐,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥_𝑂,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅   𝜑,𝑟   𝐼,𝑏,𝑐   𝐿,𝑏,𝑟   𝑅,𝑐

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝,𝑏,𝑐,𝑙,𝑥_𝑂)   𝑅(𝑥,𝑦,𝑝,𝑏,𝑥_𝑂)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏,𝑐,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,𝑝,𝑐,𝑥_𝑂)

Proof of Theorem precsexlem9

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

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . . 6 ⊢ (𝑖 = ∅ → (𝐿‘𝑖) = (𝐿‘∅))
2	1	raleqdv 3326	. . . . 5 ⊢ (𝑖 = ∅ → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ))
3		fveq2 6906	. . . . . 6 ⊢ (𝑖 = ∅ → (𝑅‘𝑖) = (𝑅‘∅))
4	3	raleqdv 3326	. . . . 5 ⊢ (𝑖 = ∅ → (∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))
5	2, 4	anbi12d 632	. . . 4 ⊢ (𝑖 = ∅ → ((∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐))))
6	5	imbi2d 340	. . 3 ⊢ (𝑖 = ∅ → ((𝜑 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))))
7		fveq2 6906	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝐿‘𝑖) = (𝐿‘𝑗))
8	7	raleqdv 3326	. . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ))
9		fveq2 6906	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑅‘𝑖) = (𝑅‘𝑗))
10	9	raleqdv 3326	. . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐)))
11	8, 10	anbi12d 632	. . . 4 ⊢ (𝑖 = 𝑗 → ((∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))))
12	11	imbi2d 340	. . 3 ⊢ (𝑖 = 𝑗 → ((𝜑 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐)))))
13		fveq2 6906	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (𝐿‘𝑖) = (𝐿‘suc 𝑗))
14	13	raleqdv 3326	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑏) <s 1s ))
15		fveq2 6906	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (𝑅‘𝑖) = (𝑅‘suc 𝑗))
16	15	raleqdv 3326	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐)))
17	14, 16	anbi12d 632	. . . . 5 ⊢ (𝑖 = suc 𝑗 → ((∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐))))
18		oveq2 7439	. . . . . . . 8 ⊢ (𝑏 = 𝑟 → (𝐴 ·s 𝑏) = (𝐴 ·s 𝑟))
19	18	breq1d 5153	. . . . . . 7 ⊢ (𝑏 = 𝑟 → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 𝑟) <s 1s ))
20	19	cbvralvw 3237	. . . . . 6 ⊢ (∀𝑏 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s )
21		oveq2 7439	. . . . . . . 8 ⊢ (𝑐 = 𝑠 → (𝐴 ·s 𝑐) = (𝐴 ·s 𝑠))
22	21	breq2d 5155	. . . . . . 7 ⊢ (𝑐 = 𝑠 → ( 1s <s (𝐴 ·s 𝑐) ↔ 1s <s (𝐴 ·s 𝑠)))
23	22	cbvralvw 3237	. . . . . 6 ⊢ (∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))
24	20, 23	anbi12i 628	. . . . 5 ⊢ ((∀𝑏 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))
25	17, 24	bitrdi 287	. . . 4 ⊢ (𝑖 = suc 𝑗 → ((∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))))
26	25	imbi2d 340	. . 3 ⊢ (𝑖 = suc 𝑗 → ((𝜑 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
27		fveq2 6906	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝐿‘𝑖) = (𝐿‘𝐼))
28	27	raleqdv 3326	. . . . 5 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ))
29		fveq2 6906	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑅‘𝑖) = (𝑅‘𝐼))
30	29	raleqdv 3326	. . . . 5 ⊢ (𝑖 = 𝐼 → (∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))
31	28, 30	anbi12d 632	. . . 4 ⊢ (𝑖 = 𝐼 → ((∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐))))
32	31	imbi2d 340	. . 3 ⊢ (𝑖 = 𝐼 → ((𝜑 → (∀𝑏 ∈ (𝐿‘𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))))
33		precsexlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
34		muls01 28138	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
35	33, 34	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
36		0slt1s 27874	. . . . . 6 ⊢ 0s <s 1s
37	35, 36	eqbrtrdi 5182	. . . . 5 ⊢ (𝜑 → (𝐴 ·s 0s ) <s 1s )
38		precsexlem.1	. . . . . . . 8 ⊢ 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 }, ∅⟩)
39		precsexlem.2	. . . . . . . 8 ⊢ 𝐿 = (1st ∘ 𝐹)
40		precsexlem.3	. . . . . . . 8 ⊢ 𝑅 = (2nd ∘ 𝐹)
41	38, 39, 40	precsexlem1 28231	. . . . . . 7 ⊢ (𝐿‘∅) = { 0s }
42	41	raleqi 3324	. . . . . 6 ⊢ (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ { 0s } (𝐴 ·s 𝑏) <s 1s )
43		0sno 27871	. . . . . . . 8 ⊢ 0s ∈ No
44	43	elexi 3503	. . . . . . 7 ⊢ 0s ∈ V
45		oveq2 7439	. . . . . . . 8 ⊢ (𝑏 = 0s → (𝐴 ·s 𝑏) = (𝐴 ·s 0s ))
46	45	breq1d 5153	. . . . . . 7 ⊢ (𝑏 = 0s → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s ))
47	44, 46	ralsn 4681	. . . . . 6 ⊢ (∀𝑏 ∈ { 0s } (𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s )
48	42, 47	bitri 275	. . . . 5 ⊢ (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s )
49	37, 48	sylibr 234	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s )
50		ral0 4513	. . . . 5 ⊢ ∀𝑐 ∈ ∅ 1s <s (𝐴 ·s 𝑐)
51	38, 39, 40	precsexlem2 28232	. . . . . 6 ⊢ (𝑅‘∅) = ∅
52	51	raleqi 3324	. . . . 5 ⊢ (∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ ∅ 1s <s (𝐴 ·s 𝑐))
53	50, 52	mpbir 231	. . . 4 ⊢ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)
54	49, 53	jctir 520	. . 3 ⊢ (𝜑 → (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))
55	38, 39, 40	precsexlem4 28234	. . . . . . . . . . . 12 ⊢ (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)})))
56	55	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (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)})))
57	56	eleq2d 2827	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) ↔ 𝑟 ∈ ((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)}))))
58		elun 4153	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ((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)})) ↔ (𝑟 ∈ (𝐿‘𝑗) ∨ 𝑟 ∈ ({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)})))
59		elun 4153	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ({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)}) ↔ (𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∨ 𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}))
60		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑟 ∈ V
61		eqeq1 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑟 → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ↔ 𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
62	61	2rexbidv 3222	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑟 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
63	60, 62	elab 3679	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅))
64		eqeq1 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑟 → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ↔ 𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
65	64	2rexbidv 3222	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑟 → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
66	60, 65	elab 3679	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)} ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))
67	63, 66	orbi12i 915	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∨ 𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
68	59, 67	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ({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)}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
69	68	orbi2i 913	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ (𝐿‘𝑗) ∨ 𝑟 ∈ ({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)})) ↔ (𝑟 ∈ (𝐿‘𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))))
70	58, 69	bitri 275	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((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)})) ↔ (𝑟 ∈ (𝐿‘𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))))
71	57, 70	bitrdi 287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) ↔ (𝑟 ∈ (𝐿‘𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))))
72		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s )
73	19	rspccv 3619	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s → (𝑟 ∈ (𝐿‘𝑗) → (𝐴 ·s 𝑟) <s 1s ))
74	72, 73	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘𝑗) → (𝐴 ·s 𝑟) <s 1s ))
75	33	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → 𝐴 ∈ No )
76	75	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
77		1sno 27872	. . . . . . . . . . . . . . . . 17 ⊢ 1s ∈ No
78	77	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s ∈ No )
79		rightssno 27920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( R ‘𝐴) ⊆ No
80	79	sseli 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ No )
81	80	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ No )
82	75	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
83	81, 82	subscld 28093	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → (𝑥𝑅 -s 𝐴) ∈ No )
84	83	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑥𝑅 -s 𝐴) ∈ No )
85		precsexlem.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0s <s 𝐴)
86		precsexlem.6	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
87	38, 39, 40, 33, 85, 86	precsexlem8 28238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐿‘𝑗) ⊆ No ∧ (𝑅‘𝑗) ⊆ No ))
88	87	simpld 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝐿‘𝑗) ⊆ No )
89	88	3adant3 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝐿‘𝑗) ⊆ No )
90	89	sselda 3983	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗)) → 𝑦𝐿 ∈ No )
91	90	adantrl 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ No )
92	84, 91	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿) ∈ No )
93	78, 92	addscld 28013	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
94	81	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ∈ No )
95	43	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s ∈ No )
96	85	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → 0s <s 𝐴)
97	96	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s 𝐴)
98		breq2 5147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝐴 <s 𝑥𝑂 ↔ 𝐴 <s 𝑥𝑅))
99		rightval 27903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( R ‘𝐴) = {𝑥𝑂 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥𝑂}
100	98, 99	elrab2 3695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) ↔ (𝑥𝑅 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝑥𝑅))
101	100	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑅)
102	101	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑥𝑅)
103	95, 82, 81, 97, 102	slttrd 27804	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s 𝑥𝑅)
104	103	sgt0ne0d 27880	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ≠ 0s )
105	104	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝑅 ≠ 0s )
106		breq2 5147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝑅))
107		oveq1 7438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑅 ·s 𝑦))
108	107	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝑅 ·s 𝑦) = 1s ))
109	108	rexbidv 3179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝑅 → (∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s ))
110	106, 109	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝑅 → (( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝑅 → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )))
111	86	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
112	111	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
113		elun2 4183	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
114	113	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
115	110, 112, 114	rspcdva 3623	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ( 0s <s 𝑥𝑅 → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s ))
116	103, 115	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )
117	116	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )
118	76, 93, 94, 105, 117	divsasswd 28228	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝑅) = (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
119		oveq2 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑦𝐿 → (𝐴 ·s 𝑏) = (𝐴 ·s 𝑦𝐿))
120	119	breq1d 5153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑦𝐿 → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 𝑦𝐿) <s 1s ))
121	120	rspccva 3621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ 𝑦𝐿 ∈ (𝐿‘𝑗)) → (𝐴 ·s 𝑦𝐿) <s 1s )
122	72, 121	sylan 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗)) → (𝐴 ·s 𝑦𝐿) <s 1s )
123	122	adantrl 716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 𝑦𝐿) <s 1s )
124	76, 91	mulscld 28161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 𝑦𝐿) ∈ No )
125	82, 81	posdifsd 28127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑥𝑅 ↔ 0s <s (𝑥𝑅 -s 𝐴)))
126	102, 125	mpbid 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s (𝑥𝑅 -s 𝐴))
127	126	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s (𝑥𝑅 -s 𝐴))
128	124, 78, 84, 127	sltmul2d 28198	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s 𝑦𝐿) <s 1s ↔ ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝑥𝑅 -s 𝐴) ·s 1s )))
129	123, 128	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝑥𝑅 -s 𝐴) ·s 1s ))
130	84	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) = (𝑥𝑅 -s 𝐴))
131	129, 130	breqtrd 5169	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s (𝑥𝑅 -s 𝐴))
132	84, 124	mulscld 28161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ∈ No )
133	76, 132, 94	sltaddsub2d 28122	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s 𝑥𝑅 ↔ ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s (𝑥𝑅 -s 𝐴)))
134	131, 133	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s 𝑥𝑅)
135	76, 78, 92	addsdid 28182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))))
136	76	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 1s ) = 𝐴)
137	76, 84, 91	muls12d 28207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) = ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
138	136, 137	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
139	135, 138	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
140	94	mulslidd 28169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s ·s 𝑥𝑅) = 𝑥𝑅)
141	134, 139, 140	3brtr4d 5175	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) <s ( 1s ·s 𝑥𝑅))
142	76, 93	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) ∈ No )
143	103	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s 𝑥𝑅)
144	142, 78, 94, 143, 117	sltdivmul2wd 28225	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝑅) <s 1s ↔ (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) <s ( 1s ·s 𝑥𝑅)))
145	141, 144	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝑅) <s 1s )
146	118, 145	eqbrtrrd 5167	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)) <s 1s )
147		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝐴 ·s 𝑟) = (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
148	147	breq1d 5153	. . . . . . . . . . . . 13 ⊢ (𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → ((𝐴 ·s 𝑟) <s 1s ↔ (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)) <s 1s ))
149	146, 148	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝐴 ·s 𝑟) <s 1s ))
150	149	rexlimdvva 3213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝐴 ·s 𝑟) <s 1s ))
151	75	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
152	77	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s ∈ No )
153		leftssno 27919	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( L ‘𝐴) ⊆ No
154		elrabi 3687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} → 𝑥𝐿 ∈ ( L ‘𝐴))
155	154	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ∈ ( L ‘𝐴))
156	153, 155	sselid 3981	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ∈ No )
157	75	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝐴 ∈ No )
158	156, 157	subscld 28093	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝑥𝐿 -s 𝐴) ∈ No )
159	158	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
160	87	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑅‘𝑗) ⊆ No )
161	160	3adant3 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑅‘𝑗) ⊆ No )
162	161	sselda 3983	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗)) → 𝑦𝑅 ∈ No )
163	162	adantrl 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ No )
164	159, 163	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅) ∈ No )
165	152, 164	addscld 28013	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
166	156	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ∈ No )
167		breq2 5147	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑥𝐿 → ( 0s <s 𝑥 ↔ 0s <s 𝑥𝐿))
168	167	elrab 3692	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ↔ (𝑥𝐿 ∈ ( L ‘𝐴) ∧ 0s <s 𝑥𝐿))
169	168	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} → 0s <s 𝑥𝐿)
170	169	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 0s <s 𝑥𝐿)
171	170	sgt0ne0d 27880	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ≠ 0s )
172	171	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝐿 ≠ 0s )
173		breq2 5147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝐿))
174		oveq1 7438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 ·s 𝑦) = (𝑥𝐿 ·s 𝑦))
175	174	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝐿 ·s 𝑦) = 1s ))
176	175	rexbidv 3179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = 𝑥𝐿 → (∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s ))
177	173, 176	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = 𝑥𝐿 → (( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝐿 → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )))
178	111	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))
179		elun1 4182	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
180	155, 179	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
181	177, 178, 180	rspcdva 3623	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ( 0s <s 𝑥𝐿 → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s ))
182	170, 181	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )
183	182	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )
184	151, 165, 166, 172, 183	divsasswd 28228	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝐿) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
185	157, 156	subscld 28093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝐴 -s 𝑥𝐿) ∈ No )
186	185	adantrr 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 -s 𝑥𝐿) ∈ No )
187	186	mulsridd 28140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) = (𝐴 -s 𝑥𝐿))
188		simp3r 1203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))
189		oveq2 7439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑦𝑅 → (𝐴 ·s 𝑐) = (𝐴 ·s 𝑦𝑅))
190	189	breq2d 5155	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑦𝑅 → ( 1s <s (𝐴 ·s 𝑐) ↔ 1s <s (𝐴 ·s 𝑦𝑅)))
191	190	rspccva 3621	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗)) → 1s <s (𝐴 ·s 𝑦𝑅))
192	188, 191	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗)) → 1s <s (𝐴 ·s 𝑦𝑅))
193	192	adantrl 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s <s (𝐴 ·s 𝑦𝑅))
194	151, 163	mulscld 28161	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s 𝑦𝑅) ∈ No )
195		breq1 5146	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 <s 𝐴 ↔ 𝑥𝐿 <s 𝐴))
196		leftval 27902	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ( L ‘𝐴) = {𝑥𝑂 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥𝑂 <s 𝐴}
197	195, 196	elrab2 3695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) ↔ (𝑥𝐿 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝑥𝐿 <s 𝐴))
198	197	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 <s 𝐴)
199	155, 198	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 <s 𝐴)
200	156, 157	posdifsd 28127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝑥𝐿 <s 𝐴 ↔ 0s <s (𝐴 -s 𝑥𝐿)))
201	199, 200	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 0s <s (𝐴 -s 𝑥𝐿))
202	201	adantrr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s (𝐴 -s 𝑥𝐿))
203	152, 194, 186, 202	sltmul2d 28198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s <s (𝐴 ·s 𝑦𝑅) ↔ ((𝐴 -s 𝑥𝐿) ·s 1s ) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅))))
204	193, 203	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
205	187, 204	eqbrtrrd 5167	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 -s 𝑥𝐿) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
206	156, 157	negsubsdi2d 28110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
207	206	adantrr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
208	159, 194	mulnegs1d 28186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝑅)) = ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
209	206	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝑅)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
210	209	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝑅)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
211	208, 210	eqtr3d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
212	205, 207, 211	3brtr4d 5175	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) <s ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
213	159, 194	mulscld 28161	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ∈ No )
214	213, 159	sltnegd 28079	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴) ↔ ( -us ‘(𝑥𝐿 -s 𝐴)) <s ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))))
215	212, 214	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴))
216	151, 213, 166	sltaddsub2d 28122	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) <s 𝑥𝐿 ↔ ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴)))
217	215, 216	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) <s 𝑥𝐿)
218	151, 152, 164	addsdid 28182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))))
219	151	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s 1s ) = 𝐴)
220	151, 159, 163	muls12d 28207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) = ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
221	219, 220	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
222	218, 221	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
223	166	mulsridd 28140	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝐿 ·s 1s ) = 𝑥𝐿)
224	217, 222, 223	3brtr4d 5175	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) <s (𝑥𝐿 ·s 1s ))
225	151, 165	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) ∈ No )
226	170	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s 𝑥𝐿)
227	225, 152, 166, 226, 183	sltdivmulwd 28224	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝐿) <s 1s ↔ (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) <s (𝑥𝐿 ·s 1s )))
228	224, 227	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝐿) <s 1s )
229	184, 228	eqbrtrrd 5167	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)) <s 1s )
230		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝐴 ·s 𝑟) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
231	230	breq1d 5153	. . . . . . . . . . . . 13 ⊢ (𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → ((𝐴 ·s 𝑟) <s 1s ↔ (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)) <s 1s ))
232	229, 231	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝐴 ·s 𝑟) <s 1s ))
233	232	rexlimdvva 3213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝐴 ·s 𝑟) <s 1s ))
234	150, 233	jaod 860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)) → (𝐴 ·s 𝑟) <s 1s ))
235	74, 234	jaod 860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ((𝑟 ∈ (𝐿‘𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))) → (𝐴 ·s 𝑟) <s 1s ))
236	71, 235	sylbid 240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) → (𝐴 ·s 𝑟) <s 1s ))
237	236	ralrimiv 3145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s )
238	38, 39, 40	precsexlem5 28235	. . . . . . . . . . . 12 ⊢ (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)})))
239	238	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (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)})))
240	239	eleq2d 2827	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) ↔ 𝑠 ∈ ((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)}))))
241		elun 4153	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((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)})) ↔ (𝑠 ∈ (𝑅‘𝑗) ∨ 𝑠 ∈ ({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)})))
242		elun 4153	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ({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)}) ↔ (𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∨ 𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))
243		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
244		eqeq1 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑠 → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ↔ 𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
245	244	2rexbidv 3222	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑠 → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
246	243, 245	elab 3679	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿))
247		eqeq1 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑠 → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ↔ 𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
248	247	2rexbidv 3222	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑠 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
249	243, 248	elab 3679	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)} ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))
250	246, 249	orbi12i 915	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∨ 𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}) ↔ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
251	242, 250	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ({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)}) ↔ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
252	251	orbi2i 913	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ (𝑅‘𝑗) ∨ 𝑠 ∈ ({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)})) ↔ (𝑠 ∈ (𝑅‘𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))))
253	241, 252	bitri 275	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((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)})) ↔ (𝑠 ∈ (𝑅‘𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))))
254	240, 253	bitrdi 287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) ↔ (𝑠 ∈ (𝑅‘𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))))
255	22	rspccv 3619	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐) → (𝑠 ∈ (𝑅‘𝑗) → 1s <s (𝐴 ·s 𝑠)))
256	188, 255	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘𝑗) → 1s <s (𝐴 ·s 𝑠)))
257	122	adantrl 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 𝑦𝐿) <s 1s )
258	75	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝐴 ∈ No )
259	90	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑦𝐿 ∈ No )
260	258, 259	mulscld 28161	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 𝑦𝐿) ∈ No )
261	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s ∈ No )
262	185	adantrr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 -s 𝑥𝐿) ∈ No )
263	201	adantrr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s (𝐴 -s 𝑥𝐿))
264	260, 261, 262, 263	sltmul2d 28198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s 𝑦𝐿) <s 1s ↔ ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝐴 -s 𝑥𝐿) ·s 1s )))
265	257, 264	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝐴 -s 𝑥𝐿) ·s 1s ))
266	262	mulsridd 28140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) = (𝐴 -s 𝑥𝐿))
267	265, 266	breqtrd 5169	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s (𝐴 -s 𝑥𝐿))
268	158	adantrr 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
269	268, 260	mulnegs1d 28186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝐿)) = ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
270	206	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝐿)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)))
271	270	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝐿)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)))
272	269, 271	eqtr3d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)))
273	206	adantrr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
274	267, 272, 273	3brtr4d 5175	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s ( -us ‘(𝑥𝐿 -s 𝐴)))
275	268, 260	mulscld 28161	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ∈ No )
276	268, 275	sltnegd 28079	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ↔ ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s ( -us ‘(𝑥𝐿 -s 𝐴))))
277	274, 276	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
278	156	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ∈ No )
279	278, 258, 275	sltsubadd2d 28120	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ↔ 𝑥𝐿 <s (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))))
280	277, 279	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 <s (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
281	278	mulslidd 28169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s ·s 𝑥𝐿) = 𝑥𝐿)
282	268, 259	mulscld 28161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿) ∈ No )
283	258, 261, 282	addsdid 28182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))))
284	258	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s 1s ) = 𝐴)
285	258, 268, 259	muls12d 28207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) = ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
286	284, 285	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
287	283, 286	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
288	280, 281, 287	3brtr4d 5175	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s ·s 𝑥𝐿) <s (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))))
289	261, 282	addscld 28013	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
290	258, 289	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) ∈ No )
291	170	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 0s <s 𝑥𝐿)
292	182	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ∃𝑦 ∈ No (𝑥𝐿 ·s 𝑦) = 1s )
293	261, 290, 278, 291, 292	sltmuldivwd 28226	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (( 1s ·s 𝑥𝐿) <s (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) ↔ 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝐿)))
294	288, 293	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝐿))
295	171	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 𝑥𝐿 ≠ 0s )
296	258, 289, 278, 295, 292	divsasswd 28228	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝐿) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
297	294, 296	breqtrd 5169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → 1s <s (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
298		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → (𝐴 ·s 𝑠) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
299	298	breq2d 5155	. . . . . . . . . . . . 13 ⊢ (𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → ( 1s <s (𝐴 ·s 𝑠) ↔ 1s <s (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿))))
300	297, 299	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿‘𝑗))) → (𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 1s <s (𝐴 ·s 𝑠)))
301	300	rexlimdvva 3213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 1s <s (𝐴 ·s 𝑠)))
302	83	adantrr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝑅 -s 𝐴) ∈ No )
303	302	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) = (𝑥𝑅 -s 𝐴))
304	192	adantrl 716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s <s (𝐴 ·s 𝑦𝑅))
305	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s ∈ No )
306	75	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝐴 ∈ No )
307	162	adantrl 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑦𝑅 ∈ No )
308	306, 307	mulscld 28161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s 𝑦𝑅) ∈ No )
309	126	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s (𝑥𝑅 -s 𝐴))
310	305, 308, 302, 309	sltmul2d 28198	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s <s (𝐴 ·s 𝑦𝑅) ↔ ((𝑥𝑅 -s 𝐴) ·s 1s ) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
311	304, 310	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
312	303, 311	eqbrtrrd 5167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑥𝑅 -s 𝐴) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
313	81	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ∈ No )
314	302, 308	mulscld 28161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ∈ No )
315	313, 306, 314	sltsubadd2d 28120	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ↔ 𝑥𝑅 <s (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))))
316	312, 315	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 <s (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
317	313	mulslidd 28169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s ·s 𝑥𝑅) = 𝑥𝑅)
318	302, 307	mulscld 28161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅) ∈ No )
319	306, 305, 318	addsdid 28182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))))
320	306	mulsridd 28140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s 1s ) = 𝐴)
321	306, 302, 307	muls12d 28207	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) = ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
322	320, 321	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
323	319, 322	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
324	316, 317, 323	3brtr4d 5175	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s ·s 𝑥𝑅) <s (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))))
325	305, 318	addscld 28013	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
326	306, 325	mulscld 28161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) ∈ No )
327	103	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 0s <s 𝑥𝑅)
328	116	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ∃𝑦 ∈ No (𝑥𝑅 ·s 𝑦) = 1s )
329	305, 326, 313, 327, 328	sltmuldivwd 28226	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (( 1s ·s 𝑥𝑅) <s (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) ↔ 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝑅)))
330	324, 329	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝑅))
331	104	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 𝑥𝑅 ≠ 0s )
332	306, 325, 313, 331, 328	divsasswd 28228	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝑅) = (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
333	330, 332	breqtrd 5169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → 1s <s (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
334		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → (𝐴 ·s 𝑠) = (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
335	334	breq2d 5155	. . . . . . . . . . . . 13 ⊢ (𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → ( 1s <s (𝐴 ·s 𝑠) ↔ 1s <s (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))))
336	333, 335	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅‘𝑗))) → (𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 1s <s (𝐴 ·s 𝑠)))
337	336	rexlimdvva 3213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 1s <s (𝐴 ·s 𝑠)))
338	301, 337	jaod 860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ((∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)) → 1s <s (𝐴 ·s 𝑠)))
339	256, 338	jaod 860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ((𝑠 ∈ (𝑅‘𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))) → 1s <s (𝐴 ·s 𝑠)))
340	254, 339	sylbid 240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) → 1s <s (𝐴 ·s 𝑠)))
341	340	ralrimiv 3145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))
342	237, 341	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))
343	342	3exp 1120	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ω → ((∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐)) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
344	343	com12 32	. . . 4 ⊢ (𝑗 ∈ ω → (𝜑 → ((∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐)) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
345	344	a2d 29	. . 3 ⊢ (𝑗 ∈ ω → ((𝜑 → (∀𝑏 ∈ (𝐿‘𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝜑 → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
346	6, 12, 26, 32, 54, 345	finds 7918	. 2 ⊢ (𝐼 ∈ ω → (𝜑 → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐))))
347	346	impcom 407	1 ⊢ ((𝜑 ∧ 𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480  ⦋csb 3899   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225   ∘ ccom 5689  suc csuc 6386  ‘cfv 6561  (class class class)co 7431  ωcom 7887  1^st c1st 8012  2^nd c2nd 8013  reccrdg 8449   No csur 27684   <s cslt 27685   bday cbday 27686   0_s c0s 27867   1_s c1s 27868   O cold 27882   L cleft 27884   R cright 27885   +_s cadds 27992   -u_s cnegs 28051   -_s csubs 28052   ·_s cmuls 28132   /_su cdivs 28213

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214

This theorem is referenced by:  precsexlem10  28240  precsexlem11  28241