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Theorem precsexlem9 28254
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 ↦ (1st𝑝) / 𝑙(2nd𝑝) / 𝑟⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
precsexlem.2 𝐿 = (1st𝐹)
precsexlem.3 𝑅 = (2nd𝐹)
precsexlem.4 (𝜑𝐴 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 ((𝜑𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐)))
Distinct variable groups:   𝐴,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥𝑂,𝑥𝐿,𝑥𝑅,𝑦,𝑦𝐿,𝑦𝑅   𝐹,𝑙,𝑝   𝐿,𝑎,𝑙,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅   𝑅,𝑎,𝑙,𝑟,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅   𝜑,𝑎,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅   𝐴,𝑏,𝑐,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥𝑂,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅   𝜑,𝑟   𝐼,𝑏,𝑐   𝐿,𝑏,𝑟   𝑅,𝑐
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝,𝑏,𝑐,𝑙,𝑥𝑂)   𝑅(𝑥,𝑦,𝑝,𝑏,𝑥𝑂)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏,𝑐,𝑥𝑂,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅)   𝐼(𝑥,𝑦,𝑟,𝑝,𝑎,𝑙,𝑥𝑂,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅)   𝐿(𝑥,𝑦,𝑝,𝑐,𝑥𝑂)

Proof of Theorem precsexlem9
Dummy variables 𝑖 𝑗 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . . 6 (𝑖 = ∅ → (𝐿𝑖) = (𝐿‘∅))
21raleqdv 3324 . . . . 5 (𝑖 = ∅ → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ))
3 fveq2 6907 . . . . . 6 (𝑖 = ∅ → (𝑅𝑖) = (𝑅‘∅))
43raleqdv 3324 . . . . 5 (𝑖 = ∅ → (∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))
52, 4anbi12d 632 . . . 4 (𝑖 = ∅ → ((∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐))))
65imbi2d 340 . . 3 (𝑖 = ∅ → ((𝜑 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))))
7 fveq2 6907 . . . . . 6 (𝑖 = 𝑗 → (𝐿𝑖) = (𝐿𝑗))
87raleqdv 3324 . . . . 5 (𝑖 = 𝑗 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ))
9 fveq2 6907 . . . . . 6 (𝑖 = 𝑗 → (𝑅𝑖) = (𝑅𝑗))
109raleqdv 3324 . . . . 5 (𝑖 = 𝑗 → (∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐)))
118, 10anbi12d 632 . . . 4 (𝑖 = 𝑗 → ((∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))))
1211imbi2d 340 . . 3 (𝑖 = 𝑗 → ((𝜑 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐)))))
13 fveq2 6907 . . . . . . 7 (𝑖 = suc 𝑗 → (𝐿𝑖) = (𝐿‘suc 𝑗))
1413raleqdv 3324 . . . . . 6 (𝑖 = suc 𝑗 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑏) <s 1s ))
15 fveq2 6907 . . . . . . 7 (𝑖 = suc 𝑗 → (𝑅𝑖) = (𝑅‘suc 𝑗))
1615raleqdv 3324 . . . . . 6 (𝑖 = suc 𝑗 → (∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐)))
1714, 16anbi12d 632 . . . . 5 (𝑖 = suc 𝑗 → ((∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 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))
1918breq1d 5158 . . . . . . 7 (𝑏 = 𝑟 → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 𝑟) <s 1s ))
2019cbvralvw 3235 . . . . . 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))
2221breq2d 5160 . . . . . . 7 (𝑐 = 𝑠 → ( 1s <s (𝐴 ·s 𝑐) ↔ 1s <s (𝐴 ·s 𝑠)))
2322cbvralvw 3235 . . . . . 6 (∀𝑐 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))
2420, 23anbi12i 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)))
2517, 24bitrdi 287 . . . 4 (𝑖 = suc 𝑗 → ((∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))))
2625imbi2d 340 . . 3 (𝑖 = suc 𝑗 → ((𝜑 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 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 6907 . . . . . 6 (𝑖 = 𝐼 → (𝐿𝑖) = (𝐿𝐼))
2827raleqdv 3324 . . . . 5 (𝑖 = 𝐼 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ))
29 fveq2 6907 . . . . . 6 (𝑖 = 𝐼 → (𝑅𝑖) = (𝑅𝐼))
3029raleqdv 3324 . . . . 5 (𝑖 = 𝐼 → (∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐)))
3128, 30anbi12d 632 . . . 4 (𝑖 = 𝐼 → ((∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐)) ↔ (∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐))))
3231imbi2d 340 . . 3 (𝑖 = 𝐼 → ((𝜑 → (∀𝑏 ∈ (𝐿𝑖)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑖) 1s <s (𝐴 ·s 𝑐))) ↔ (𝜑 → (∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐)))))
33 precsexlem.4 . . . . . . 7 (𝜑𝐴 No )
34 muls01 28153 . . . . . . 7 (𝐴 No → (𝐴 ·s 0s ) = 0s )
3533, 34syl 17 . . . . . 6 (𝜑 → (𝐴 ·s 0s ) = 0s )
36 0slt1s 27889 . . . . . 6 0s <s 1s
3735, 36eqbrtrdi 5187 . . . . 5 (𝜑 → (𝐴 ·s 0s ) <s 1s )
38 precsexlem.1 . . . . . . . 8 𝐹 = rec((𝑝 ∈ V ↦ (1st𝑝) / 𝑙(2nd𝑝) / 𝑟⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
39 precsexlem.2 . . . . . . . 8 𝐿 = (1st𝐹)
40 precsexlem.3 . . . . . . . 8 𝑅 = (2nd𝐹)
4138, 39, 40precsexlem1 28246 . . . . . . 7 (𝐿‘∅) = { 0s }
4241raleqi 3322 . . . . . 6 (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ↔ ∀𝑏 ∈ { 0s } (𝐴 ·s 𝑏) <s 1s )
43 0sno 27886 . . . . . . . 8 0s No
4443elexi 3501 . . . . . . 7 0s ∈ V
45 oveq2 7439 . . . . . . . 8 (𝑏 = 0s → (𝐴 ·s 𝑏) = (𝐴 ·s 0s ))
4645breq1d 5158 . . . . . . 7 (𝑏 = 0s → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s ))
4744, 46ralsn 4686 . . . . . 6 (∀𝑏 ∈ { 0s } (𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s )
4842, 47bitri 275 . . . . 5 (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 0s ) <s 1s )
4937, 48sylibr 234 . . . 4 (𝜑 → ∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s )
50 ral0 4519 . . . . 5 𝑐 ∈ ∅ 1s <s (𝐴 ·s 𝑐)
5138, 39, 40precsexlem2 28247 . . . . . 6 (𝑅‘∅) = ∅
5251raleqi 3322 . . . . 5 (∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐) ↔ ∀𝑐 ∈ ∅ 1s <s (𝐴 ·s 𝑐))
5350, 52mpbir 231 . . . 4 𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)
5449, 53jctir 520 . . 3 (𝜑 → (∀𝑏 ∈ (𝐿‘∅)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘∅) 1s <s (𝐴 ·s 𝑐)))
5538, 39, 40precsexlem4 28249 . . . . . . . . . . . 12 (𝑗 ∈ ω → (𝐿‘suc 𝑗) = ((𝐿𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
56553ad2ant2 1133 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝐿‘suc 𝑗) = ((𝐿𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
5756eleq2d 2825 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) ↔ 𝑟 ∈ ((𝐿𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}))))
58 elun 4163 . . . . . . . . . . 11 (𝑟 ∈ ((𝐿𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})) ↔ (𝑟 ∈ (𝐿𝑗) ∨ 𝑟 ∈ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
59 elun 4163 . . . . . . . . . . . . 13 (𝑟 ∈ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}) ↔ (𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∨ 𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}))
60 vex 3482 . . . . . . . . . . . . . . 15 𝑟 ∈ V
61 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑟 → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ↔ 𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
62612rexbidv 3220 . . . . . . . . . . . . . . 15 (𝑎 = 𝑟 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)))
6360, 62elab 3681 . . . . . . . . . . . . . 14 (𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅))
64 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑟 → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ↔ 𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
65642rexbidv 3220 . . . . . . . . . . . . . . 15 (𝑎 = 𝑟 → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
6660, 65elab 3681 . . . . . . . . . . . . . 14 (𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)} ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))
6763, 66orbi12i 914 . . . . . . . . . . . . 13 ((𝑟 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∨ 𝑟 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
6859, 67bitri 275 . . . . . . . . . . . 12 (𝑟 ∈ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
6968orbi2i 912 . . . . . . . . . . 11 ((𝑟 ∈ (𝐿𝑗) ∨ 𝑟 ∈ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})) ↔ (𝑟 ∈ (𝐿𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))))
7058, 69bitri 275 . . . . . . . . . 10 (𝑟 ∈ ((𝐿𝑗) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})) ↔ (𝑟 ∈ (𝐿𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))))
7157, 70bitrdi 287 . . . . . . . . 9 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) ↔ (𝑟 ∈ (𝐿𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))))
72 simp3l 1200 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s )
7319rspccv 3619 . . . . . . . . . . 11 (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s → (𝑟 ∈ (𝐿𝑗) → (𝐴 ·s 𝑟) <s 1s ))
7472, 73syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿𝑗) → (𝐴 ·s 𝑟) <s 1s ))
75333ad2ant1 1132 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → 𝐴 No )
7675adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝐴 No )
77 1sno 27887 . . . . . . . . . . . . . . . . 17 1s No
7877a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 1s No )
79 rightssno 27935 . . . . . . . . . . . . . . . . . . . . 21 ( R ‘𝐴) ⊆ No
8079sseli 3991 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 No )
8180adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 No )
8275adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝐴 No )
8381, 82subscld 28108 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → (𝑥𝑅 -s 𝐴) ∈ No )
8483adantrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (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 ))
8738, 39, 40, 33, 85, 86precsexlem8 28253 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ω) → ((𝐿𝑗) ⊆ No ∧ (𝑅𝑗) ⊆ No ))
8887simpld 494 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ω) → (𝐿𝑗) ⊆ No )
89883adant3 1131 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝐿𝑗) ⊆ No )
9089sselda 3995 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝐿 ∈ (𝐿𝑗)) → 𝑦𝐿 No )
9190adantrl 716 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑦𝐿 No )
9284, 91mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿) ∈ No )
9378, 92addscld 28028 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
9481adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑥𝑅 No )
9543a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s No )
96853ad2ant1 1132 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → 0s <s 𝐴)
9796adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s 𝐴)
98 breq2 5152 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑥𝑅 → (𝐴 <s 𝑥𝑂𝐴 <s 𝑥𝑅))
99 rightval 27918 . . . . . . . . . . . . . . . . . . . . 21 ( R ‘𝐴) = {𝑥𝑂 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥𝑂}
10098, 99elrab2 3698 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑅 ∈ ( R ‘𝐴) ↔ (𝑥𝑅 ∈ ( O ‘( bday 𝐴)) ∧ 𝐴 <s 𝑥𝑅))
101100simprbi 496 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑅 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑅)
102101adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑥𝑅)
10395, 82, 81, 97, 102slttrd 27819 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s 𝑥𝑅)
104103sgt0ne0d 27895 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ≠ 0s )
105104adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑥𝑅 ≠ 0s )
106 breq2 5152 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑂 = 𝑥𝑅 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝑅))
107 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑥𝑅 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑅 ·s 𝑦))
108107eqeq1d 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑂 = 𝑥𝑅 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝑅 ·s 𝑦) = 1s ))
109108rexbidv 3177 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑂 = 𝑥𝑅 → (∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s ))
110106, 109imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑥𝑂 = 𝑥𝑅 → (( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝑅 → ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s )))
111863ad2ant1 1132 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ))
112111adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ))
113 elun2 4193 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
114113adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
115110, 112, 114rspcdva 3623 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ( 0s <s 𝑥𝑅 → ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s ))
116103, 115mpd 15 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s )
117116adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s )
11876, 93, 94, 105, 117divsasswd 28243 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((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))
120119breq1d 5158 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑦𝐿 → ((𝐴 ·s 𝑏) <s 1s ↔ (𝐴 ·s 𝑦𝐿) <s 1s ))
121120rspccva 3621 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s𝑦𝐿 ∈ (𝐿𝑗)) → (𝐴 ·s 𝑦𝐿) <s 1s )
12272, 121sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝐿 ∈ (𝐿𝑗)) → (𝐴 ·s 𝑦𝐿) <s 1s )
123122adantrl 716 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 𝑦𝐿) <s 1s )
12476, 91mulscld 28176 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 𝑦𝐿) ∈ No )
12582, 81posdifsd 28142 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑥𝑅 ↔ 0s <s (𝑥𝑅 -s 𝐴)))
126102, 125mpbid 232 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝑅 ∈ ( R ‘𝐴)) → 0s <s (𝑥𝑅 -s 𝐴))
127126adantrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 0s <s (𝑥𝑅 -s 𝐴))
128124, 78, 84, 127sltmul2d 28213 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s 𝑦𝐿) <s 1s ↔ ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝑥𝑅 -s 𝐴) ·s 1s )))
129123, 128mpbid 232 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝑥𝑅 -s 𝐴) ·s 1s ))
13084mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) = (𝑥𝑅 -s 𝐴))
131129, 130breqtrd 5174 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s (𝑥𝑅 -s 𝐴))
13284, 124mulscld 28176 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ∈ No )
13376, 132, 94sltaddsub2d 28137 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s 𝑥𝑅 ↔ ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) <s (𝑥𝑅 -s 𝐴)))
134131, 133mpbird 257 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s 𝑥𝑅)
13576, 78, 92addsdid 28197 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))))
13676mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 1s ) = 𝐴)
13776, 84, 91muls12d 28222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) = ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
138136, 137oveq12d 7449 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
139135, 138eqtrd 2775 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
14094mulslidd 28184 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( 1s ·s 𝑥𝑅) = 𝑥𝑅)
141134, 139, 1403brtr4d 5180 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) <s ( 1s ·s 𝑥𝑅))
14276, 93mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) ∈ No )
143103adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 0s <s 𝑥𝑅)
144142, 78, 94, 143, 117sltdivmul2wd 28240 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (((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)))
145141, 144mpbird 257 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝑅) <s 1s )
146118, 145eqbrtrrd 5172 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (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)))
148147breq1d 5158 . . . . . . . . . . . . 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 ))
149146, 148syl5ibrcom 247 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝐴 ·s 𝑟) <s 1s ))
150149rexlimdvva 3211 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) → (𝐴 ·s 𝑟) <s 1s ))
15175adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝐴 No )
15277a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 1s No )
153 leftssno 27934 . . . . . . . . . . . . . . . . . . . 20 ( L ‘𝐴) ⊆ No
154 elrabi 3690 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} → 𝑥𝐿 ∈ ( L ‘𝐴))
155154adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ∈ ( L ‘𝐴))
156153, 155sselid 3993 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 No )
15775adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝐴 No )
158156, 157subscld 28108 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝑥𝐿 -s 𝐴) ∈ No )
159158adantrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
16087simprd 495 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ω) → (𝑅𝑗) ⊆ No )
1611603adant3 1131 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑅𝑗) ⊆ No )
162161sselda 3995 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝑅 ∈ (𝑅𝑗)) → 𝑦𝑅 No )
163162adantrl 716 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑦𝑅 No )
164159, 163mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅) ∈ No )
165152, 164addscld 28028 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
166156adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑥𝐿 No )
167 breq2 5152 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑥𝐿 → ( 0s <s 𝑥 ↔ 0s <s 𝑥𝐿))
168167elrab 3695 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ↔ (𝑥𝐿 ∈ ( L ‘𝐴) ∧ 0s <s 𝑥𝐿))
169168simprbi 496 . . . . . . . . . . . . . . . . . 18 (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} → 0s <s 𝑥𝐿)
170169adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 0s <s 𝑥𝐿)
171170sgt0ne0d 27895 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ≠ 0s )
172171adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑥𝐿 ≠ 0s )
173 breq2 5152 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑂 = 𝑥𝐿 → ( 0s <s 𝑥𝑂 ↔ 0s <s 𝑥𝐿))
174 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 ·s 𝑦) = (𝑥𝐿 ·s 𝑦))
175174eqeq1d 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑂 = 𝑥𝐿 → ((𝑥𝑂 ·s 𝑦) = 1s ↔ (𝑥𝐿 ·s 𝑦) = 1s ))
176175rexbidv 3177 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑂 = 𝑥𝐿 → (∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ↔ ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s ))
177173, 176imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑥𝑂 = 𝑥𝐿 → (( 0s <s 𝑥𝑂 → ∃𝑦 No (𝑥𝑂 ·s 𝑦) = 1s ) ↔ ( 0s <s 𝑥𝐿 → ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s )))
178111adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 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 4192 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
180155, 179syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
181177, 178, 180rspcdva 3623 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ( 0s <s 𝑥𝐿 → ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s ))
182170, 181mpd 15 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s )
183182adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s )
184151, 165, 166, 172, 183divsasswd 28243 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝐿) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)))
185157, 156subscld 28108 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝐴 -s 𝑥𝐿) ∈ No )
186185adantrr 717 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 -s 𝑥𝐿) ∈ No )
187186mulsridd 28155 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) = (𝐴 -s 𝑥𝐿))
188 simp3r 1201 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))
189 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑦𝑅 → (𝐴 ·s 𝑐) = (𝐴 ·s 𝑦𝑅))
190189breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑦𝑅 → ( 1s <s (𝐴 ·s 𝑐) ↔ 1s <s (𝐴 ·s 𝑦𝑅)))
191190rspccva 3621 . . . . . . . . . . . . . . . . . . . . . . 23 ((∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐) ∧ 𝑦𝑅 ∈ (𝑅𝑗)) → 1s <s (𝐴 ·s 𝑦𝑅))
192188, 191sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑦𝑅 ∈ (𝑅𝑗)) → 1s <s (𝐴 ·s 𝑦𝑅))
193192adantrl 716 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 1s <s (𝐴 ·s 𝑦𝑅))
194151, 163mulscld 28176 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s 𝑦𝑅) ∈ No )
195 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝑂 = 𝑥𝐿 → (𝑥𝑂 <s 𝐴𝑥𝐿 <s 𝐴))
196 leftval 27917 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ( L ‘𝐴) = {𝑥𝑂 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥𝑂 <s 𝐴}
197195, 196elrab2 3698 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝐿 ∈ ( L ‘𝐴) ↔ (𝑥𝐿 ∈ ( O ‘( bday 𝐴)) ∧ 𝑥𝐿 <s 𝐴))
198197simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 <s 𝐴)
199155, 198syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 𝑥𝐿 <s 𝐴)
200156, 157posdifsd 28142 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → (𝑥𝐿 <s 𝐴 ↔ 0s <s (𝐴 -s 𝑥𝐿)))
201199, 200mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → 0s <s (𝐴 -s 𝑥𝐿))
202201adantrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 0s <s (𝐴 -s 𝑥𝐿))
203152, 194, 186, 202sltmul2d 28213 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s <s (𝐴 ·s 𝑦𝑅) ↔ ((𝐴 -s 𝑥𝐿) ·s 1s ) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅))))
204193, 203mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
205187, 204eqbrtrrd 5172 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 -s 𝑥𝐿) <s ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
206156, 157negsubsdi2d 28125 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ 𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
207206adantrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
208159, 194mulnegs1d 28201 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝑅)) = ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
209206oveq1d 7446 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 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)))
210209adantrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝑅)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
211208, 210eqtr3d 2777 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝑅)))
212205, 207, 2113brtr4d 5180 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) <s ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
213159, 194mulscld 28176 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ∈ No )
214213, 159sltnegd 28094 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴) ↔ ( -us ‘(𝑥𝐿 -s 𝐴)) <s ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))))
215212, 214mpbird 257 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴))
216151, 213, 166sltaddsub2d 28137 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) <s 𝑥𝐿 ↔ ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) <s (𝑥𝐿 -s 𝐴)))
217215, 216mpbird 257 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))) <s 𝑥𝐿)
218151, 152, 164addsdid 28197 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))))
219151mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s 1s ) = 𝐴)
220151, 159, 163muls12d 28222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) = ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
221219, 220oveq12d 7449 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
222218, 221eqtrd 2775 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
223166mulsridd 28155 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑥𝐿 ·s 1s ) = 𝑥𝐿)
224217, 222, 2233brtr4d 5180 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) <s (𝑥𝐿 ·s 1s ))
225151, 165mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) ∈ No )
226170adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 0s <s 𝑥𝐿)
227225, 152, 166, 226, 183sltdivmulwd 28239 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (((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 )))
228224, 227mpbird 257 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝐿) <s 1s )
229184, 228eqbrtrrd 5172 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (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)))
231230breq1d 5158 . . . . . . . . . . . . 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 ))
232229, 231syl5ibrcom 247 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝐴 ·s 𝑟) <s 1s ))
233232rexlimdvva 3211 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿) → (𝐴 ·s 𝑟) <s 1s ))
234150, 233jaod 859 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)) → (𝐴 ·s 𝑟) <s 1s ))
23574, 234jaod 859 . . . . . . . . 9 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ((𝑟 ∈ (𝐿𝑗) ∨ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿𝑗)𝑟 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅) ∨ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅𝑗)𝑟 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿))) → (𝐴 ·s 𝑟) <s 1s ))
23671, 235sylbid 240 . . . . . . . 8 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑟 ∈ (𝐿‘suc 𝑗) → (𝐴 ·s 𝑟) <s 1s ))
237236ralrimiv 3143 . . . . . . 7 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s )
23838, 39, 40precsexlem5 28250 . . . . . . . . . . . 12 (𝑗 ∈ ω → (𝑅‘suc 𝑗) = ((𝑅𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
2392383ad2ant2 1133 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑅‘suc 𝑗) = ((𝑅𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
240239eleq2d 2825 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) ↔ 𝑠 ∈ ((𝑅𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))))
241 elun 4163 . . . . . . . . . . 11 (𝑠 ∈ ((𝑅𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})) ↔ (𝑠 ∈ (𝑅𝑗) ∨ 𝑠 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
242 elun 4163 . . . . . . . . . . . . 13 (𝑠 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}) ↔ (𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∨ 𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))
243 vex 3482 . . . . . . . . . . . . . . 15 𝑠 ∈ V
244 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑠 → (𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ↔ 𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
2452442rexbidv 3220 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
246243, 245elab 3681 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ↔ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿))
247 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑠 → (𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ↔ 𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
2482472rexbidv 3220 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
249243, 248elab 3681 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)} ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))
250246, 249orbi12i 914 . . . . . . . . . . . . 13 ((𝑠 ∈ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∨ 𝑠 ∈ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}) ↔ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
251242, 250bitri 275 . . . . . . . . . . . 12 (𝑠 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}) ↔ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
252251orbi2i 912 . . . . . . . . . . 11 ((𝑠 ∈ (𝑅𝑗) ∨ 𝑠 ∈ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})) ↔ (𝑠 ∈ (𝑅𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))))
253241, 252bitri 275 . . . . . . . . . 10 (𝑠 ∈ ((𝑅𝑗) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})) ↔ (𝑠 ∈ (𝑅𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))))
254240, 253bitrdi 287 . . . . . . . . 9 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) ↔ (𝑠 ∈ (𝑅𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))))
25522rspccv 3619 . . . . . . . . . . 11 (∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐) → (𝑠 ∈ (𝑅𝑗) → 1s <s (𝐴 ·s 𝑠)))
256188, 255syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅𝑗) → 1s <s (𝐴 ·s 𝑠)))
257122adantrl 716 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 𝑦𝐿) <s 1s )
25875adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝐴 No )
25990adantrl 716 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑦𝐿 No )
260258, 259mulscld 28176 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 𝑦𝐿) ∈ No )
26177a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 1s No )
262185adantrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 -s 𝑥𝐿) ∈ No )
263201adantrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 0s <s (𝐴 -s 𝑥𝐿))
264260, 261, 262, 263sltmul2d 28213 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s 𝑦𝐿) <s 1s ↔ ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝐴 -s 𝑥𝐿) ·s 1s )))
265257, 264mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s ((𝐴 -s 𝑥𝐿) ·s 1s ))
266262mulsridd 28155 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 -s 𝑥𝐿) ·s 1s ) = (𝐴 -s 𝑥𝐿))
267265, 266breqtrd 5174 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)) <s (𝐴 -s 𝑥𝐿))
268158adantrr 717 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝑥𝐿 -s 𝐴) ∈ No )
269268, 260mulnegs1d 28201 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝐿)) = ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
270206oveq1d 7446 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 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)))
271270adantrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (( -us ‘(𝑥𝐿 -s 𝐴)) ·s (𝐴 ·s 𝑦𝐿)) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)))
272269, 271eqtr3d 2777 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) = ((𝐴 -s 𝑥𝐿) ·s (𝐴 ·s 𝑦𝐿)))
273206adantrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( -us ‘(𝑥𝐿 -s 𝐴)) = (𝐴 -s 𝑥𝐿))
274267, 272, 2733brtr4d 5180 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s ( -us ‘(𝑥𝐿 -s 𝐴)))
275268, 260mulscld 28176 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ∈ No )
276268, 275sltnegd 28094 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ↔ ( -us ‘((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))) <s ( -us ‘(𝑥𝐿 -s 𝐴))))
277274, 276mpbird 257 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
278156adantrr 717 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑥𝐿 No )
279278, 258, 275sltsubadd2d 28135 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝐿 -s 𝐴) <s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)) ↔ 𝑥𝐿 <s (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))))
280277, 279mpbid 232 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑥𝐿 <s (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
281278mulslidd 28184 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( 1s ·s 𝑥𝐿) = 𝑥𝐿)
282268, 259mulscld 28176 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿) ∈ No )
283258, 261, 282addsdid 28197 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))))
284258mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s 1s ) = 𝐴)
285258, 268, 259muls12d 28222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) = ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿)))
286284, 285oveq12d 7449 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
287283, 286eqtrd 2775 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) = (𝐴 +s ((𝑥𝐿 -s 𝐴) ·s (𝐴 ·s 𝑦𝐿))))
288280, 281, 2873brtr4d 5180 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( 1s ·s 𝑥𝐿) <s (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))))
289261, 282addscld 28028 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) ∈ No )
290258, 289mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) ∈ No )
291170adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 0s <s 𝑥𝐿)
292182adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ∃𝑦 No (𝑥𝐿 ·s 𝑦) = 1s )
293261, 290, 278, 291, 292sltmuldivwd 28241 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (( 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)))
294288, 293mpbid 232 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝐿))
295171adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 𝑥𝐿 ≠ 0s )
296258, 289, 278, 295, 292divsasswd 28243 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿))) /su 𝑥𝐿) = (𝐴 ·s (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)))
297294, 296breqtrd 5174 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → 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)))
299298breq2d 5160 . . . . . . . . . . . . 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))))
300297, 299syl5ibrcom 247 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥} ∧ 𝑦𝐿 ∈ (𝐿𝑗))) → (𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 1s <s (𝐴 ·s 𝑠)))
301300rexlimdvva 3211 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) → 1s <s (𝐴 ·s 𝑠)))
30283adantrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑥𝑅 -s 𝐴) ∈ No )
303302mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) = (𝑥𝑅 -s 𝐴))
304192adantrl 716 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 1s <s (𝐴 ·s 𝑦𝑅))
30577a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 1s No )
30675adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝐴 No )
307162adantrl 716 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑦𝑅 No )
308306, 307mulscld 28176 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s 𝑦𝑅) ∈ No )
309126adantrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 0s <s (𝑥𝑅 -s 𝐴))
310305, 308, 302, 309sltmul2d 28213 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s <s (𝐴 ·s 𝑦𝑅) ↔ ((𝑥𝑅 -s 𝐴) ·s 1s ) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
311304, 310mpbid 232 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 1s ) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
312303, 311eqbrtrrd 5172 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑥𝑅 -s 𝐴) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
31381adantrr 717 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑥𝑅 No )
314302, 308mulscld 28176 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ∈ No )
315313, 306, 314sltsubadd2d 28135 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝑅 -s 𝐴) <s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)) ↔ 𝑥𝑅 <s (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))))
316312, 315mpbid 232 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑥𝑅 <s (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
317313mulslidd 28184 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s ·s 𝑥𝑅) = 𝑥𝑅)
318302, 307mulscld 28176 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅) ∈ No )
319306, 305, 318addsdid 28197 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))))
320306mulsridd 28155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s 1s ) = 𝐴)
321306, 302, 307muls12d 28222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) = ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅)))
322320, 321oveq12d 7449 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 ·s 1s ) +s (𝐴 ·s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
323319, 322eqtrd 2775 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) = (𝐴 +s ((𝑥𝑅 -s 𝐴) ·s (𝐴 ·s 𝑦𝑅))))
324316, 317, 3233brtr4d 5180 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s ·s 𝑥𝑅) <s (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))))
325305, 318addscld 28028 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) ∈ No )
326306, 325mulscld 28176 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) ∈ No )
327103adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 0s <s 𝑥𝑅)
328116adantrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ∃𝑦 No (𝑥𝑅 ·s 𝑦) = 1s )
329305, 326, 313, 327, 328sltmuldivwd 28241 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (( 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)))
330324, 329mpbid 232 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 1s <s ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝑅))
331104adantrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 𝑥𝑅 ≠ 0s )
332306, 325, 313, 331, 328divsasswd 28243 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → ((𝐴 ·s ( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅))) /su 𝑥𝑅) = (𝐴 ·s (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)))
333330, 332breqtrd 5174 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → 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)))
335334breq2d 5160 . . . . . . . . . . . . 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))))
336333, 335syl5ibrcom 247 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑦𝑅 ∈ (𝑅𝑗))) → (𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 1s <s (𝐴 ·s 𝑠)))
337336rexlimdvva 3211 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅) → 1s <s (𝐴 ·s 𝑠)))
338301, 337jaod 859 . . . . . . . . . 10 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ((∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)) → 1s <s (𝐴 ·s 𝑠)))
339256, 338jaod 859 . . . . . . . . 9 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ((𝑠 ∈ (𝑅𝑗) ∨ (∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿𝑗)𝑠 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅𝑗)𝑠 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅))) → 1s <s (𝐴 ·s 𝑠)))
340254, 339sylbid 240 . . . . . . . 8 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝑠 ∈ (𝑅‘suc 𝑗) → 1s <s (𝐴 ·s 𝑠)))
341340ralrimiv 3143 . . . . . . 7 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠))
342237, 341jca 511 . . . . . 6 ((𝜑𝑗 ∈ ω ∧ (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))
3433423exp 1118 . . . . 5 (𝜑 → (𝑗 ∈ ω → ((∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐)) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
344343com12 32 . . . 4 (𝑗 ∈ ω → (𝜑 → ((∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐)) → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
345344a2d 29 . . 3 (𝑗 ∈ ω → ((𝜑 → (∀𝑏 ∈ (𝐿𝑗)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝑗) 1s <s (𝐴 ·s 𝑐))) → (𝜑 → (∀𝑟 ∈ (𝐿‘suc 𝑗)(𝐴 ·s 𝑟) <s 1s ∧ ∀𝑠 ∈ (𝑅‘suc 𝑗) 1s <s (𝐴 ·s 𝑠)))))
3466, 12, 26, 32, 54, 345finds 7919 . 2 (𝐼 ∈ ω → (𝜑 → (∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐))))
347346impcom 407 1 ((𝜑𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅𝐼) 1s <s (𝐴 ·s 𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231  ccom 5693  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  2nd c2nd 8012  reccrdg 8448   No csur 27699   <s cslt 27700   bday cbday 27701   0s c0s 27882   1s c1s 27883   O cold 27897   L cleft 27899   R cright 27900   +s cadds 28007   -us cnegs 28066   -s csubs 28067   ·s cmuls 28147   /su cdivs 28228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229
This theorem is referenced by:  precsexlem10  28255  precsexlem11  28256
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