ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemcbv GIF version

Theorem caucvgprprlemcbv 6842
Description: Lemma for caucvgprpr 6867. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
Assertion
Ref Expression
caucvgprprlemcbv (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
Distinct variable groups:   𝐹,𝑎,𝑏,𝑘   𝑛,𝐹,𝑎,𝑘   𝑎,𝑙,𝑏,𝑘   𝑢,𝑎,𝑏,𝑘   𝑛,𝑙   𝑢,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑎,𝑏,𝑙)   𝐹(𝑢,𝑙)

Proof of Theorem caucvgprprlemcbv
StepHypRef Expression
1 caucvgprpr.cau . 2 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
2 breq1 3794 . . . 4 (𝑛 = 𝑎 → (𝑛 <N 𝑘𝑎 <N 𝑘))
3 fveq2 5205 . . . . . 6 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
4 opeq1 3576 . . . . . . . . . . . 12 (𝑛 = 𝑎 → ⟨𝑛, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
54eceq1d 6172 . . . . . . . . . . 11 (𝑛 = 𝑎 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
65fveq2d 5209 . . . . . . . . . 10 (𝑛 = 𝑎 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
76breq2d 3803 . . . . . . . . 9 (𝑛 = 𝑎 → (𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
87abbidv 2171 . . . . . . . 8 (𝑛 = 𝑎 → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )})
96breq1d 3801 . . . . . . . . 9 (𝑛 = 𝑎 → ((*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢))
109abbidv 2171 . . . . . . . 8 (𝑛 = 𝑎 → {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢})
118, 10opeq12d 3584 . . . . . . 7 (𝑛 = 𝑎 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
1211oveq2d 5555 . . . . . 6 (𝑛 = 𝑎 → ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
133, 12breq12d 3804 . . . . 5 (𝑛 = 𝑎 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))
143, 11oveq12d 5557 . . . . . 6 (𝑛 = 𝑎 → ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
1514breq2d 3803 . . . . 5 (𝑛 = 𝑎 → ((𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))
1613, 15anbi12d 450 . . . 4 (𝑛 = 𝑎 → (((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
172, 16imbi12d 227 . . 3 (𝑛 = 𝑎 → ((𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝑎 <N 𝑘 → ((𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))))
18 breq2 3795 . . . 4 (𝑘 = 𝑏 → (𝑎 <N 𝑘𝑎 <N 𝑏))
19 fveq2 5205 . . . . . . 7 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
2019oveq1d 5554 . . . . . 6 (𝑘 = 𝑏 → ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
2120breq2d 3803 . . . . 5 (𝑘 = 𝑏 → ((𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))
2219breq1d 3801 . . . . 5 (𝑘 = 𝑏 → ((𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))
2321, 22anbi12d 450 . . . 4 (𝑘 = 𝑏 → (((𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
2418, 23imbi12d 227 . . 3 (𝑘 = 𝑏 → ((𝑎 <N 𝑘 → ((𝐹𝑎)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)))))
2517, 24cbvral2v 2558 . 2 (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
261, 25sylib 131 1 (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  {cab 2042  wral 2323  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fv 4937  df-ov 5542  df-ec 6138
This theorem is referenced by:  caucvgprprlemval  6843
  Copyright terms: Public domain W3C validator