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Theorem caucvgprlemloc 6646
 Description: Lemma for caucvgpr 6653. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f (φ𝐹:NQ)
caucvgpr.cau (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (φ𝑗 N A <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
Assertion
Ref Expression
caucvgprlemloc (φ𝑠 Q 𝑟 Q (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 6392 . . . . 5 (𝑠 <Q 𝑟y Q (𝑠 +Q y) = 𝑟)
21adantl 262 . . . 4 (((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) → y Q (𝑠 +Q y) = 𝑟)
3 subhalfnqq 6397 . . . . . 6 (y Qx Q (x +Q x) <Q y)
43ad2antrl 459 . . . . 5 ((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) → x Q (x +Q x) <Q y)
5 archrecnq 6635 . . . . . . 7 (x Q𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)
65ad2antrl 459 . . . . . 6 (((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) → 𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)
7 simprr 484 . . . . . . . . . . . 12 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)
8 nnnq 6405 . . . . . . . . . . . . . . 15 (𝑚 N → [⟨𝑚, 1𝑜⟩] ~Q Q)
9 recclnq 6376 . . . . . . . . . . . . . . 15 ([⟨𝑚, 1𝑜⟩] ~Q Q → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q)
108, 9syl 14 . . . . . . . . . . . . . 14 (𝑚 N → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q)
1110ad2antrl 459 . . . . . . . . . . . . 13 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q)
12 simplrl 487 . . . . . . . . . . . . 13 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → x Q)
13 lt2addnq 6388 . . . . . . . . . . . . 13 ((((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q x Q) ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q x Q)) → (((*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 1135 . . . . . . . . . . . 12 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 409 . . . . . . . . . . 11 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (x +Q x))
16 simplrr 488 . . . . . . . . . . 11 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (x +Q x) <Q y)
17 ltsonq 6382 . . . . . . . . . . . 12 <Q Or Q
18 ltrelnq 6349 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
1917, 18sotri 4663 . . . . . . . . . . 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 391 . . . . . . . . . 10 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q y)
21 simplrl 487 . . . . . . . . . . 11 (((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) → 𝑠 Q)
2221ad3antrrr 461 . . . . . . . . . 10 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → 𝑠 Q)
23 ltanqi 6386 . . . . . . . . . 10 ((((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q y 𝑠 Q) → (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q (𝑠 +Q y))
2420, 22, 23syl2anc 391 . . . . . . . . 9 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q (𝑠 +Q y))
25 simprr 484 . . . . . . . . . 10 ((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) → (𝑠 +Q y) = 𝑟)
2625ad2antrr 457 . . . . . . . . 9 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝑠 +Q y) = 𝑟)
2724, 26breqtrd 3779 . . . . . . . 8 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q 𝑟)
28 addclnq 6359 . . . . . . . . . . 11 (((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q) → ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
2911, 11, 28syl2anc 391 . . . . . . . . . 10 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
30 addclnq 6359 . . . . . . . . . 10 ((𝑠 Q ((*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 391 . . . . . . . . 9 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) Q)
32 simplrr 488 . . . . . . . . . 10 (((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) → 𝑟 Q)
3332ad3antrrr 461 . . . . . . . . 9 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → 𝑟 Q)
34 caucvgpr.f . . . . . . . . . . . 12 (φ𝐹:NQ)
3534ad5antr 465 . . . . . . . . . . 11 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → 𝐹:NQ)
36 simprl 483 . . . . . . . . . . 11 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → 𝑚 N)
3735, 36ffvelrnd 5246 . . . . . . . . . 10 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝐹𝑚) Q)
38 addclnq 6359 . . . . . . . . . 10 (((𝐹𝑚) Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
3937, 11, 38syl2anc 391 . . . . . . . . 9 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
40 sowlin 4048 . . . . . . . . . 10 (( <Q Or Q ((𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) Q 𝑟 Q ((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 400 . . . . . . . . 9 (((𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) Q 𝑟 Q ((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 1134 . . . . . . . 8 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 261 . . . . . . . . . 10 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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)
45 simplrl 487 . . . . . . . . . . 11 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) → 𝑚 N)
46 simpr 103 . . . . . . . . . . . . 13 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 261 . . . . . . . . . . . . . . 15 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 6366 . . . . . . . . . . . . . . 15 ((𝑠 Q (*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 1134 . . . . . . . . . . . . . 14 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 3765 . . . . . . . . . . . . 13 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 156 . . . . . . . . . . . 12 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 6384 . . . . . . . . . . . . . 14 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
5352adantl 262 . . . . . . . . . . . . 13 ((((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
54 addclnq 6359 . . . . . . . . . . . . . 14 ((𝑠 Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q) → (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
5544, 47, 54syl2anc 391 . . . . . . . . . . . . 13 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 261 . . . . . . . . . . . . 13 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 6365 . . . . . . . . . . . . . 14 ((f Q g Q) → (f +Q g) = (g +Q f))
5857adantl 262 . . . . . . . . . . . . 13 ((((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) (f Q g Q)) → (f +Q g) = (g +Q f))
5953, 55, 56, 47, 58caovord2d 5612 . . . . . . . . . . . 12 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 156 . . . . . . . . . . 11 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 3540 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
6261eceq1d 6078 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
6362fveq2d 5125 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
6463oveq2d 5471 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
65 fveq2 5121 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
6664, 65breq12d 3768 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (𝐹𝑚)))
6766rspcev 2650 . . . . . . . . . . 11 ((𝑚 N (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (𝐹𝑚)) → 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
6845, 60, 67syl2anc 391 . . . . . . . . . 10 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) (𝑠 +Q ((*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) <Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))) → 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
69 oveq1 5462 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
7069breq1d 3765 . . . . . . . . . . . 12 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
7170rexbidv 2321 . . . . . . . . . . 11 (𝑙 = 𝑠 → (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
72 caucvgpr.lim . . . . . . . . . . . . 13 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
7372fveq2i 5124 . . . . . . . . . . . 12 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
74 nqex 6347 . . . . . . . . . . . . . 14 Q V
7574rabex 3892 . . . . . . . . . . . . 13 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} V
7674rabex 3892 . . . . . . . . . . . . 13 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} V
7775, 76op1st 5715 . . . . . . . . . . . 12 (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
7873, 77eqtri 2057 . . . . . . . . . . 11 (1st𝐿) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
7971, 78elrab2 2694 . . . . . . . . . 10 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
8044, 68, 79sylanbrc 394 . . . . . . . . 9 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 108 . . . . . . . 8 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 261 . . . . . . . . . 10 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟) → 𝑟 Q)
8365, 63oveq12d 5473 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
8483breq1d 3765 . . . . . . . . . . . 12 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟))
8584rspcev 2650 . . . . . . . . . . 11 ((𝑚 N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
8636, 85sylan 267 . . . . . . . . . 10 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
87 breq2 3759 . . . . . . . . . . . 12 (u = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
8887rexbidv 2321 . . . . . . . . . . 11 (u = 𝑟 → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
8972fveq2i 5124 . . . . . . . . . . . 12 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
9075, 76op2nd 5716 . . . . . . . . . . . 12 (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
9189, 90eqtri 2057 . . . . . . . . . . 11 (2nd𝐿) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
9288, 91elrab2 2694 . . . . . . . . . 10 (𝑟 (2nd𝐿) ↔ (𝑟 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
9382, 86, 92sylanbrc 394 . . . . . . . . 9 (((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟) → 𝑟 (2nd𝐿))
9493ex 108 . . . . . . . 8 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟𝑟 (2nd𝐿)))
9581, 94orim12d 699 . . . . . . 7 ((((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*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 (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) (𝑚 N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q x)) → (𝑠 (1st𝐿) 𝑟 (2nd𝐿)))
976, 96rexlimddv 2431 . . . . 5 (((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) (x Q (x +Q x) <Q y)) → (𝑠 (1st𝐿) 𝑟 (2nd𝐿)))
984, 97rexlimddv 2431 . . . 4 ((((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) (y Q (𝑠 +Q y) = 𝑟)) → (𝑠 (1st𝐿) 𝑟 (2nd𝐿)))
992, 98rexlimddv 2431 . . 3 (((φ (𝑠 Q 𝑟 Q)) 𝑠 <Q 𝑟) → (𝑠 (1st𝐿) 𝑟 (2nd𝐿)))
10099ex 108 . 2 ((φ (𝑠 Q 𝑟 Q)) → (𝑠 <Q 𝑟 → (𝑠 (1st𝐿) 𝑟 (2nd𝐿))))
101100ralrimivva 2395 1 (φ𝑠 Q 𝑟 Q (𝑠 <Q 𝑟 → (𝑠 (1st𝐿) 𝑟 (2nd𝐿))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755   Or wor 4023  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  1𝑜c1o 5933  [cec 6040  Ncnpi 6256
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