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Theorem caucvgprlemloc 7137
Description: Lemma for caucvgpr 7144. 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 6871 . . . . 5 (𝑠 <Q 𝑟 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑟)
21adantl 271 . . . 4 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑟)
3 subhalfnqq 6876 . . . . . 6 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
43ad2antrl 474 . . . . 5 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
5 archrecnq 7125 . . . . . . 7 (𝑥Q → ∃𝑚N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q 𝑥)
65ad2antrl 474 . . . . . 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 499 . . . . . . . . . . . 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 6884 . . . . . . . . . . . . . . 15 (𝑚N → [⟨𝑚, 1𝑜⟩] ~QQ)
9 recclnq 6854 . . . . . . . . . . . . . . 15 ([⟨𝑚, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q)
108, 9syl 14 . . . . . . . . . . . . . 14 (𝑚N → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q)
1110ad2antrl 474 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 6866 . . . . . . . . . . . . 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 1171 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 503 . . . . . . . . . . 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 6860 . . . . . . . . . . . 12 <Q Or Q
18 ltrelnq 6827 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
1917, 18sotri 4782 . . . . . . . . . . 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 403 . . . . . . . . . 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 502 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → 𝑠Q)
2221ad3antrrr 476 . . . . . . . . . 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 6864 . . . . . . . . . 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 403 . . . . . . . . 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 499 . . . . . . . . . 10 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 +Q 𝑦) = 𝑟)
2625ad2antrr 472 . . . . . . . . 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 3835 . . . . . . . 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 6837 . . . . . . . . . . 11 (((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
2911, 11, 28syl2anc 403 . . . . . . . . . 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 6837 . . . . . . . . . 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 403 . . . . . . . . 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 503 . . . . . . . . . 10 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → 𝑟Q)
3332ad3antrrr 476 . . . . . . . . 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 480 . . . . . . . . . . 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 498 . . . . . . . . . . 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, 36ffvelrnd 5380 . . . . . . . . . 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 6837 . . . . . . . . . 10 (((𝐹𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
3937, 11, 38syl2anc 403 . . . . . . . . 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 4111 . . . . . . . . . 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 415 . . . . . . . . 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 1170 . . . . . . . 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 270 . . . . . . . . . 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 502 . . . . . . . . . . 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 108 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . . 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 3821 . . . . . . . . . . . . 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 165 . . . . . . . . . . . 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 6862 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
5352adantl 271 . . . . . . . . . . . . 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 6837 . . . . . . . . . . . . . 14 ((𝑠Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
5544, 47, 54syl2anc 403 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5857adantl 271 . . . . . . . . . . . . 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 5749 . . . . . . . . . . . 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 165 . . . . . . . . . . 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 3596 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
6261eceq1d 6258 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
6362fveq2d 5257 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
6463oveq2d 5607 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
65 fveq2 5253 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
6664, 65breq12d 3824 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (𝐹𝑚)))
6766rspcev 2712 . . . . . . . . . . 11 ((𝑚N ∧ (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (𝐹𝑚)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
6845, 60, 67syl2anc 403 . . . . . . . . . 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 5598 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
7069breq1d 3821 . . . . . . . . . . . 12 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
7170rexbidv 2375 . . . . . . . . . . 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 5256 . . . . . . . . . . . 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 6825 . . . . . . . . . . . . . 14 Q ∈ V
7574rabex 3948 . . . . . . . . . . . . 13 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
7674rabex 3948 . . . . . . . . . . . . 13 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
7775, 76op1st 5852 . . . . . . . . . . . 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 2103 . . . . . . . . . . 11 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
7971, 78elrab2 2762 . . . . . . . . . 10 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
8044, 68, 79sylanbrc 408 . . . . . . . . 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 113 . . . . . . . 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 270 . . . . . . . . . 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 5609 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
8483breq1d 3821 . . . . . . . . . . . 12 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟))
8584rspcev 2712 . . . . . . . . . . 11 ((𝑚N ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
8636, 85sylan 277 . . . . . . . . . 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 3815 . . . . . . . . . . . 12 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
8887rexbidv 2375 . . . . . . . . . . 11 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
8972fveq2i 5256 . . . . . . . . . . . 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 5853 . . . . . . . . . . . 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 2103 . . . . . . . . . . 11 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
9288, 91elrab2 2762 . . . . . . . . . 10 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
9382, 86, 92sylanbrc 408 . . . . . . . . 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 113 . . . . . . . 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 733 . . . . . . 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 2487 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
984, 97rexlimddv 2487 . . . 4 ((((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
992, 98rexlimddv 2487 . . 3 (((𝜑 ∧ (𝑠Q𝑟Q)) ∧ 𝑠 <Q 𝑟) → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))
10099ex 113 . 2 ((𝜑 ∧ (𝑠Q𝑟Q)) → (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
101100ralrimivva 2449 1 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  cop 3425   class class class wbr 3811   Or wor 4086  wf 4965  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  1𝑜c1o 6106  [cec 6220  Ncnpi 6734   <N clti 6737   ~Q ceq 6741  Qcnq 6742   +Q cplq 6744  *Qcrq 6746   <Q cltq 6747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815
This theorem is referenced by:  caucvgprlemcl  7138
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