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Theorem caucvgprlemloc 7687
Description: Lemma for caucvgpr 7694. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-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
caucvgprlemloc (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙   𝑢,𝐹   𝜑,𝑗,𝑟,𝑠   𝑠,𝑙   𝑢,𝑗,𝑟
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑘,𝑛,𝑠,𝑟)   𝐿(𝑢,𝑗,𝑘,𝑛,𝑠,𝑟,𝑙)

Proof of Theorem caucvgprlemloc
Dummy variables 𝑓 𝑔 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 7421 . . . . 5 (𝑠 <Q 𝑟 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑟)
21adantl 277 . . . 4 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑟)
3 subhalfnqq 7426 . . . . . 6 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
43ad2antrl 490 . . . . 5 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
5 archrecnq 7675 . . . . . . 7 (𝑥Q → ∃𝑚N (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
65ad2antrl 490 . . . . . 6 (((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → ∃𝑚N (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
7 simprr 531 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
8 nnnq 7434 . . . . . . . . . . . . . . 15 (𝑚N → [⟨𝑚, 1o⟩] ~QQ)
9 recclnq 7404 . . . . . . . . . . . . . . 15 ([⟨𝑚, 1o⟩] ~QQ → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
108, 9syl 14 . . . . . . . . . . . . . 14 (𝑚N → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
1110ad2antrl 490 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
12 simplrl 535 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑥Q)
13 lt2addnq 7416 . . . . . . . . . . . . 13 ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q𝑥Q) ∧ ((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q𝑥Q)) → (((*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥 ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥)))
1411, 12, 11, 12, 13syl22anc 1249 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥 ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥)))
157, 7, 14mp2and 433 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥))
16 simplrr 536 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑥 +Q 𝑥) <Q 𝑦)
17 ltsonq 7410 . . . . . . . . . . . 12 <Q Or Q
18 ltrelnq 7377 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
1917, 18sotri 5036 . . . . . . . . . . 11 ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥) ∧ (𝑥 +Q 𝑥) <Q 𝑦) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦)
2015, 16, 19syl2anc 411 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦)
21 simplrl 535 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → 𝑠Q)
2221ad3antrrr 492 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑠Q)
23 ltanqi 7414 . . . . . . . . . 10 ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦𝑠Q) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q (𝑠 +Q 𝑦))
2420, 22, 23syl2anc 411 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q (𝑠 +Q 𝑦))
25 simprr 531 . . . . . . . . . 10 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 +Q 𝑦) = 𝑟)
2625ad2antrr 488 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q 𝑦) = 𝑟)
2724, 26breqtrd 4041 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟)
28 addclnq 7387 . . . . . . . . . . 11 (((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
2911, 11, 28syl2anc 411 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
30 addclnq 7387 . . . . . . . . . 10 ((𝑠Q ∧ ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q)
3122, 29, 30syl2anc 411 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q)
32 simplrr 536 . . . . . . . . . 10 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → 𝑟Q)
3332ad3antrrr 492 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑟Q)
34 caucvgpr.f . . . . . . . . . . . 12 (𝜑𝐹:NQ)
3534ad5antr 496 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝐹:NQ)
36 simprl 529 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑚N)
3735, 36ffvelcdmd 5665 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝐹𝑚) ∈ Q)
38 addclnq 7387 . . . . . . . . . 10 (((𝐹𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
3937, 11, 38syl2anc 411 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
40 sowlin 4332 . . . . . . . . . 10 (( <Q Or Q ∧ ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q𝑟Q ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
4117, 40mpan 424 . . . . . . . . 9 (((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q𝑟Q ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
4231, 33, 39, 41syl3anc 1248 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
4327, 42mpd 13 . . . . . . 7 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟))
4422adantr 276 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑠Q)
45 simplrl 535 . . . . . . . . . . 11 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑚N)
46 simpr 110 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
4711adantr 276 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
48 addassnqg 7394 . . . . . . . . . . . . . . 15 ((𝑠Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) = (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
4944, 47, 47, 48syl3anc 1248 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) = (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
5049breq1d 4025 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ↔ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
5146, 50mpbird 167 . . . . . . . . . . . 12 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
52 ltanqg 7412 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
5352adantl 277 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
54 addclnq 7387 . . . . . . . . . . . . . 14 ((𝑠Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
5544, 47, 54syl2anc 411 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
5637adantr 276 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝐹𝑚) ∈ Q)
57 addcomnqg 7393 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5857adantl 277 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5953, 55, 56, 47, 58caovord2d 6057 . . . . . . . . . . . 12 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹𝑚) ↔ ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
6051, 59mpbird 167 . . . . . . . . . . 11 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹𝑚))
61 opeq1 3790 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ⟨𝑗, 1o⟩ = ⟨𝑚, 1o⟩)
6261eceq1d 6584 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑚, 1o⟩] ~Q )
6362fveq2d 5531 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑚, 1o⟩] ~Q ))
6463oveq2d 5904 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
65 fveq2 5527 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
6664, 65breq12d 4028 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹𝑚)))
6766rspcev 2853 . . . . . . . . . . 11 ((𝑚N ∧ (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹𝑚)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
6845, 60, 67syl2anc 411 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
69 oveq1 5895 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
7069breq1d 4025 . . . . . . . . . . . 12 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
7170rexbidv 2488 . . . . . . . . . . 11 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
72 caucvgpr.lim . . . . . . . . . . . . 13 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
7372fveq2i 5530 . . . . . . . . . . . 12 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
74 nqex 7375 . . . . . . . . . . . . . 14 Q ∈ V
7574rabex 4159 . . . . . . . . . . . . 13 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
7674rabex 4159 . . . . . . . . . . . . 13 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
7775, 76op1st 6160 . . . . . . . . . . . 12 (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
7873, 77eqtri 2208 . . . . . . . . . . 11 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
7971, 78elrab2 2908 . . . . . . . . . 10 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
8044, 68, 79sylanbrc 417 . . . . . . . . 9 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑠 ∈ (1st𝐿))
8180ex 115 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) → 𝑠 ∈ (1st𝐿)))
8233adantr 276 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → 𝑟Q)
8365, 63oveq12d 5906 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
8483breq1d 4025 . . . . . . . . . . . 12 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟))
8584rspcev 2853 . . . . . . . . . . 11 ((𝑚N ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
8636, 85sylan 283 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
87 breq2 4019 . . . . . . . . . . . 12 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
8887rexbidv 2488 . . . . . . . . . . 11 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
8972fveq2i 5530 . . . . . . . . . . . 12 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
9075, 76op2nd 6161 . . . . . . . . . . . 12 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
9189, 90eqtri 2208 . . . . . . . . . . 11 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
9288, 91elrab2 2908 . . . . . . . . . 10 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
9382, 86, 92sylanbrc 417 . . . . . . . . 9 (((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → 𝑟 ∈ (2nd𝐿))
9493ex 115 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟𝑟 ∈ (2nd𝐿)))
9581, 94orim12d 787 . . . . . . 7 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
9643, 95mpd 13 . . . . . 6 ((((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
976, 96rexlimddv 2609 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
984, 97rexlimddv 2609 . . . 4 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
992, 98rexlimddv 2609 . . 3 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
10099ex 115 . 2 ((𝜑 ∧ (𝑠Q𝑟Q)) → (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
101100ralrimivva 2569 1 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  w3a 979   = wceq 1363  wcel 2158  wral 2465  wrex 2466  {crab 2469  cop 3607   class class class wbr 4015   Or wor 4307  wf 5224  cfv 5228  (class class class)co 5888  1st c1st 6152  2nd c2nd 6153  1oc1o 6423  [cec 6546  Ncnpi 7284   <N clti 7287   ~Q ceq 7291  Qcnq 7292   +Q cplq 7294  *Qcrq 7296   <Q cltq 7297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365
This theorem is referenced by:  caucvgprlemcl  7688
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