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Theorem cauappcvgprlemlim 6633
Description: Lemma for cauappcvgpr 6634. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (φ𝐹:QQ)
cauappcvgpr.app (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (φ𝑝 Q A <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
Assertion
Ref Expression
cauappcvgprlemlim (φ𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
Distinct variable groups:   A,𝑝   𝐿,𝑝,𝑞   φ,𝑝,𝑞   𝐹,𝑙,𝑝,𝑞,𝑟,u   𝐿,𝑟
Allowed substitution hints:   φ(u,𝑟,𝑙)   A(u,𝑟,𝑞,𝑙)   𝐿(u,𝑙)

Proof of Theorem cauappcvgprlemlim
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6 (φ𝐹:QQ)
21adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝐹:QQ)
3 cauappcvgpr.app . . . . . 6 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
43adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
5 cauappcvgpr.bnd . . . . . 6 (φ𝑝 Q A <Q (𝐹𝑝))
65adantr 261 . . . . 5 ((φ (x Q y Q)) → 𝑝 Q A <Q (𝐹𝑝))
7 cauappcvgpr.lim . . . . 5 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
8 simprl 483 . . . . 5 ((φ (x Q y Q)) → x Q)
9 simprr 484 . . . . 5 ((φ (x Q y Q)) → y Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 6631 . . . 4 ((φ (x Q y Q)) → ⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩))
112, 4, 6, 7, 8, 9cauappcvgprlem2 6632 . . . 4 ((φ (x Q y Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩)
1210, 11jca 290 . . 3 ((φ (x Q y Q)) → (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩))
1312ralrimivva 2395 . 2 (φx Q y Q (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩))
14 fveq2 5121 . . . . . . . 8 (x = 𝑞 → (𝐹x) = (𝐹𝑞))
1514breq2d 3767 . . . . . . 7 (x = 𝑞 → (𝑙 <Q (𝐹x) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2152 . . . . . 6 (x = 𝑞 → {𝑙𝑙 <Q (𝐹x)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 3765 . . . . . . 7 (x = 𝑞 → ((𝐹x) <Q u ↔ (𝐹𝑞) <Q u))
1817abbidv 2152 . . . . . 6 (x = 𝑞 → {u ∣ (𝐹x) <Q u} = {u ∣ (𝐹𝑞) <Q u})
1916, 18opeq12d 3548 . . . . 5 (x = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩)
20 oveq1 5462 . . . . . . . . 9 (x = 𝑞 → (x +Q y) = (𝑞 +Q y))
2120breq2d 3767 . . . . . . . 8 (x = 𝑞 → (𝑙 <Q (x +Q y) ↔ 𝑙 <Q (𝑞 +Q y)))
2221abbidv 2152 . . . . . . 7 (x = 𝑞 → {𝑙𝑙 <Q (x +Q y)} = {𝑙𝑙 <Q (𝑞 +Q y)})
2320breq1d 3765 . . . . . . . 8 (x = 𝑞 → ((x +Q y) <Q u ↔ (𝑞 +Q y) <Q u))
2423abbidv 2152 . . . . . . 7 (x = 𝑞 → {u ∣ (x +Q y) <Q u} = {u ∣ (𝑞 +Q y) <Q u})
2522, 24opeq12d 3548 . . . . . 6 (x = 𝑞 → ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩)
2625oveq2d 5471 . . . . 5 (x = 𝑞 → (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩))
2719, 26breq12d 3768 . . . 4 (x = 𝑞 → (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩)))
2814, 20oveq12d 5473 . . . . . . . 8 (x = 𝑞 → ((𝐹x) +Q (x +Q y)) = ((𝐹𝑞) +Q (𝑞 +Q y)))
2928breq2d 3767 . . . . . . 7 (x = 𝑞 → (𝑙 <Q ((𝐹x) +Q (x +Q y)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))))
3029abbidv 2152 . . . . . 6 (x = 𝑞 → {𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))})
3128breq1d 3765 . . . . . . 7 (x = 𝑞 → (((𝐹x) +Q (x +Q y)) <Q u ↔ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u))
3231abbidv 2152 . . . . . 6 (x = 𝑞 → {u ∣ ((𝐹x) +Q (x +Q y)) <Q u} = {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u})
3330, 32opeq12d 3548 . . . . 5 (x = 𝑞 → ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩)
3433breq2d 3767 . . . 4 (x = 𝑞 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩))
3527, 34anbi12d 442 . . 3 (x = 𝑞 → ((⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩)))
36 oveq2 5463 . . . . . . . . 9 (y = 𝑟 → (𝑞 +Q y) = (𝑞 +Q 𝑟))
3736breq2d 3767 . . . . . . . 8 (y = 𝑟 → (𝑙 <Q (𝑞 +Q y) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2152 . . . . . . 7 (y = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q y)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 3765 . . . . . . . 8 (y = 𝑟 → ((𝑞 +Q y) <Q u ↔ (𝑞 +Q 𝑟) <Q u))
4039abbidv 2152 . . . . . . 7 (y = 𝑟 → {u ∣ (𝑞 +Q y) <Q u} = {u ∣ (𝑞 +Q 𝑟) <Q u})
4138, 40opeq12d 3548 . . . . . 6 (y = 𝑟 → ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩)
4241oveq2d 5471 . . . . 5 (y = 𝑟 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩))
4342breq2d 3767 . . . 4 (y = 𝑟 → (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩)))
4436oveq2d 5471 . . . . . . . 8 (y = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q y)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 3767 . . . . . . 7 (y = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2152 . . . . . 6 (y = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 3765 . . . . . . 7 (y = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q y)) <Q u ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u))
4847abbidv 2152 . . . . . 6 (y = 𝑟 → {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u} = {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u})
4946, 48opeq12d 3548 . . . . 5 (y = 𝑟 → ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩)
5049breq2d 3767 . . . 4 (y = 𝑟 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
5143, 50anbi12d 442 . . 3 (y = 𝑟 → ((⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q y)}, {u ∣ (𝑞 +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q y))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q y)) <Q u}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩)))
5235, 51cbvral2v 2535 . 2 (x Q y Q (⟨{𝑙𝑙 <Q (𝐹x)}, {u ∣ (𝐹x) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (x +Q y)}, {u ∣ (x +Q y) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹x) +Q (x +Q y))}, {u ∣ ((𝐹x) +Q (x +Q y)) <Q u}⟩) ↔ 𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
5313, 52sylib 127 1 (φ𝑞 Q 𝑟 Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {u ∣ (𝐹𝑞) <Q u}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {u ∣ (𝑞 +Q 𝑟) <Q u}⟩) 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {u ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q u}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  wf 4841  cfv 4845  (class class class)co 5455  Qcnq 6264   +Q cplq 6266   <Q cltq 6269   +P cpp 6277  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-iltp 6453
This theorem is referenced by:  cauappcvgpr  6634
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