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 Description: Lemma for caucvgpr 6986. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemladdrl (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙   𝑛,𝐹,𝑘   𝑘,𝐿,𝑗   𝑆,𝑙,𝑢,𝑗   𝑗,𝑘,𝑆
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑆(𝑛)   𝐿(𝑢,𝑛,𝑙)

Dummy variables 𝑟 𝑓 𝑔 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3590 . . . . . . . . 9 (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
21eceq1d 6229 . . . . . . . 8 (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
32fveq2d 5233 . . . . . . 7 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
43oveq2d 5579 . . . . . 6 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
5 fveq2 5229 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
65oveq1d 5578 . . . . . 6 (𝑗 = 𝑎 → ((𝐹𝑗) +Q 𝑆) = ((𝐹𝑎) +Q 𝑆))
74, 6breq12d 3818 . . . . 5 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
87cbvrexv 2583 . . . 4 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆))
98a1i 9 . . 3 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
109rabbiia 2596 . 2 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)}
11 oveq1 5570 . . . . . . 7 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1211breq1d 3815 . . . . . 6 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) ↔ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
1312rexbidv 2374 . . . . 5 (𝑙 = 𝑟 → (∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) ↔ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
1413elrab 2757 . . . 4 (𝑟 ∈ {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} ↔ (𝑟Q ∧ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)))
15 caucvgpr.f . . . . . . . . . . . . . . 15 (𝜑𝐹:NQ)
1615ad4antr 478 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝐹:NQ)
17 caucvgpr.cau . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
1817ad4antr 478 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
19 simpr 108 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑏N)
20 simpllr 501 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑎N)
2116, 18, 19, 20caucvgprlemnbj 6971 . . . . . . . . . . . . 13 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ¬ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎))
2215ad3antrrr 476 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝐹:NQ)
2322ffvelrnda 5354 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝐹𝑏) ∈ Q)
24 nnnq 6726 . . . . . . . . . . . . . . . . . 18 (𝑏N → [⟨𝑏, 1𝑜⟩] ~QQ)
25 recclnq 6696 . . . . . . . . . . . . . . . . . 18 ([⟨𝑏, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
2619, 24, 253syl 17 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q)
27 addclnq 6679 . . . . . . . . . . . . . . . . 17 (((𝐹𝑏) ∈ Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
2823, 26, 27syl2anc 403 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
29 nnnq 6726 . . . . . . . . . . . . . . . . 17 (𝑎N → [⟨𝑎, 1𝑜⟩] ~QQ)
30 recclnq 6696 . . . . . . . . . . . . . . . . 17 ([⟨𝑎, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q)
3120, 29, 303syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q)
32 caucvgprlemladd.s . . . . . . . . . . . . . . . . 17 (𝜑𝑆Q)
3332ad4antr 478 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → 𝑆Q)
34 addassnqg 6686 . . . . . . . . . . . . . . . 16 ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q𝑆Q) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) +Q 𝑆) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆)))
3528, 31, 33, 34syl3anc 1170 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) +Q 𝑆) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆)))
3635breq1d 3815 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) +Q 𝑆) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆)) <Q ((𝐹𝑎) +Q 𝑆)))
37 ltanqg 6704 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3837adantl 271 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
39 addclnq 6679 . . . . . . . . . . . . . . . 16 ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ Q)
4028, 31, 39syl2anc 403 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ Q)
4116, 20ffvelrnd 5355 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝐹𝑎) ∈ Q)
42 addcomnqg 6685 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4342adantl 271 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4438, 40, 41, 33, 43caovord2d 5721 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎) ↔ ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) +Q 𝑆) <Q ((𝐹𝑎) +Q 𝑆)))
45 addcomnqg 6685 . . . . . . . . . . . . . . . . 17 ((𝑆Q ∧ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆))
4633, 31, 45syl2anc 403 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆))
4746oveq2d 5579 . . . . . . . . . . . . . . 15 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) = (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆)))
4847breq1d 3815 . . . . . . . . . . . . . 14 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q ((*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) +Q 𝑆)) <Q ((𝐹𝑎) +Q 𝑆)))
4936, 44, 483bitr4rd 219 . . . . . . . . . . . . 13 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆) ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
5021, 49mtbird 631 . . . . . . . . . . . 12 (((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) ∧ 𝑏N) → ¬ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))
5150nrexdv 2459 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))
5251intnand 874 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
5317ad3antrrr 476 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
54 caucvgpr.bnd . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
55 fveq2 5229 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑏 → (𝐹𝑗) = (𝐹𝑏))
5655breq2d 3817 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑏 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑏)))
5756cbvralv 2582 . . . . . . . . . . . . . . 15 (∀𝑗N 𝐴 <Q (𝐹𝑗) ↔ ∀𝑏N 𝐴 <Q (𝐹𝑏))
5854, 57sylib 120 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑏N 𝐴 <Q (𝐹𝑏))
5958ad3antrrr 476 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ∀𝑏N 𝐴 <Q (𝐹𝑏))
60 caucvgpr.lim . . . . . . . . . . . . . 14 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
61 opeq1 3590 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑏 → ⟨𝑗, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
6261eceq1d 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑏 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
6362fveq2d 5233 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑏 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
6463oveq2d 5579 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑏 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
6564, 55breq12d 3818 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑏 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏)))
6665cbvrexv 2583 . . . . . . . . . . . . . . . . 17 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏))
6766a1i 9 . . . . . . . . . . . . . . . 16 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏)))
6867rabbiia 2596 . . . . . . . . . . . . . . 15 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏)}
6955, 63oveq12d 5581 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑏 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
7069breq1d 3815 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑏 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢))
7170cbvrexv 2583 . . . . . . . . . . . . . . . . 17 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢)
7271a1i 9 . . . . . . . . . . . . . . . 16 (𝑢Q → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢))
7372rabbiia 2596 . . . . . . . . . . . . . . 15 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢}
7468, 73opeq12i 3595 . . . . . . . . . . . . . 14 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏)}, {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
7560, 74eqtri 2103 . . . . . . . . . . . . 13 𝐿 = ⟨{𝑙Q ∣ ∃𝑏N (𝑙 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q (𝐹𝑏)}, {𝑢Q ∣ ∃𝑏N ((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
7632ad3antrrr 476 . . . . . . . . . . . . . 14 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑆Q)
7729, 30syl 14 . . . . . . . . . . . . . . 15 (𝑎N → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q)
7877ad2antlr 473 . . . . . . . . . . . . . 14 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q)
79 addclnq 6679 . . . . . . . . . . . . . 14 ((𝑆Q ∧ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ Q)
8076, 78, 79syl2anc 403 . . . . . . . . . . . . 13 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ Q)
8122, 53, 59, 75, 80caucvgprlemladdfu 6981 . . . . . . . . . . . 12 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q 𝑢})
8281sseld 3007 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) → ((𝐹𝑎) +Q 𝑆) ∈ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q 𝑢}))
83 breq2 3809 . . . . . . . . . . . . 13 (𝑢 = ((𝐹𝑎) +Q 𝑆) → ((((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q 𝑢 ↔ (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8483rexbidv 2374 . . . . . . . . . . . 12 (𝑢 = ((𝐹𝑎) +Q 𝑆) → (∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q 𝑢 ↔ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8584elrab 2757 . . . . . . . . . . 11 (((𝐹𝑎) +Q 𝑆) ∈ {𝑢Q ∣ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q 𝑢} ↔ (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆)))
8682, 85syl6ib 159 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) → (((𝐹𝑎) +Q 𝑆) ∈ Q ∧ ∃𝑏N (((𝐹𝑏) +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) +Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑎) +Q 𝑆))))
8752, 86mtod 622 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ¬ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)))
8815, 17, 54, 60caucvgprlemcl 6980 . . . . . . . . . . . 12 (𝜑𝐿P)
8988ad3antrrr 476 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝐿P)
90 nqprlu 6851 . . . . . . . . . . . 12 ((𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ Q → ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
9180, 90syl 14 . . . . . . . . . . 11 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
92 addclpr 6841 . . . . . . . . . . 11 ((𝐿P ∧ ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
9389, 91, 92syl2anc 403 . . . . . . . . . 10 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) ∈ P)
94 prop 6779 . . . . . . . . . . 11 ((𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P)
95 prloc 6795 . . . . . . . . . . 11 ((⟨(1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)), (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩))⟩ ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩))))
9694, 95sylan 277 . . . . . . . . . 10 (((𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) ∈ P ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩))))
9793, 96sylancom 411 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) ∨ ((𝐹𝑎) +Q 𝑆) ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩))))
9887, 97ecased 1281 . . . . . . . 8 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)))
99 simpllr 501 . . . . . . . . 9 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟Q)
10089, 76, 99, 78caucvgprlemcanl 6948 . . . . . . . 8 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → ((𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)) ↔ 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
10198, 100mpbid 145 . . . . . . 7 ((((𝜑𝑟Q) ∧ 𝑎N) ∧ (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
102101ex 113 . . . . . 6 (((𝜑𝑟Q) ∧ 𝑎N) → ((𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
103102rexlimdva 2482 . . . . 5 ((𝜑𝑟Q) → (∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
104103expimpd 355 . . . 4 (𝜑 → ((𝑟Q ∧ ∃𝑎N (𝑟 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)) → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
10514, 104syl5bi 150 . . 3 (𝜑 → (𝑟 ∈ {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} → 𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
106105ssrdv 3014 . 2 (𝜑 → {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q ((𝐹𝑎) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
10710, 106syl5eqss 3052 1 (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  {cab 2069  ∀wral 2353  ∃wrex 2354  {crab 2357   ⊆ wss 2982  ⟨cop 3419   class class class wbr 3805  ⟶wf 4948  ‘cfv 4952  (class class class)co 5563  1st c1st 5816  2nd c2nd 5817  1𝑜c1o 6078  [cec 6191  Ncnpi 6576
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