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Theorem climeqmpt 40447
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
climeqmpt.x 𝑥𝜑
climeqmpt.a (𝜑𝐴𝑉)
climeqmpt.b (𝜑𝐵𝑊)
climeqmpt.m (𝜑𝑀 ∈ ℤ)
climeqmpt.z 𝑍 = (ℤ𝑀)
climeqmpt.s (𝜑𝑍𝐴)
climeqmpt.t (𝜑𝑍𝐵)
climeqmpt.c ((𝜑𝑥𝑍) → 𝐶𝑈)
Assertion
Ref Expression
climeqmpt (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑈(𝑥)   𝑀(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem climeqmpt
StepHypRef Expression
1 climeqmpt.x . 2 𝑥𝜑
2 nfmpt1 4881 . 2 𝑥(𝑥𝐴𝐶)
3 nfmpt1 4881 . 2 𝑥(𝑥𝐵𝐶)
4 climeqmpt.m . 2 (𝜑𝑀 ∈ ℤ)
5 climeqmpt.z . 2 𝑍 = (ℤ𝑀)
6 climeqmpt.a . . 3 (𝜑𝐴𝑉)
76mptexd 6631 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ V)
8 climeqmpt.b . . 3 (𝜑𝐵𝑊)
98mptexd 6631 . 2 (𝜑 → (𝑥𝐵𝐶) ∈ V)
10 climeqmpt.s . . . . . 6 (𝜑𝑍𝐴)
1110adantr 466 . . . . 5 ((𝜑𝑥𝑍) → 𝑍𝐴)
12 simpr 471 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝑍)
1311, 12sseldd 3753 . . . 4 ((𝜑𝑥𝑍) → 𝑥𝐴)
14 climeqmpt.c . . . 4 ((𝜑𝑥𝑍) → 𝐶𝑈)
15 eqid 2771 . . . . 5 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1615fvmpt2 6433 . . . 4 ((𝑥𝐴𝐶𝑈) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1713, 14, 16syl2anc 573 . . 3 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
18 climeqmpt.t . . . . . . 7 (𝜑𝑍𝐵)
1918adantr 466 . . . . . 6 ((𝜑𝑥𝑍) → 𝑍𝐵)
2019, 12sseldd 3753 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝐵)
21 eqid 2771 . . . . . 6 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
2221fvmpt2 6433 . . . . 5 ((𝑥𝐵𝐶𝑈) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2320, 14, 22syl2anc 573 . . . 4 ((𝜑𝑥𝑍) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2423eqcomd 2777 . . 3 ((𝜑𝑥𝑍) → 𝐶 = ((𝑥𝐵𝐶)‘𝑥))
2517, 24eqtrd 2805 . 2 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐵𝐶)‘𝑥))
261, 2, 3, 4, 5, 7, 9, 25climeqf 40438 1 (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  wcel 2145  Vcvv 3351  wss 3723   class class class wbr 4786  cmpt 4863  cfv 6031  cz 11579  cuz 11888  cli 14423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-neg 10471  df-z 11580  df-uz 11889  df-clim 14427
This theorem is referenced by:  smflimsuplem6  41551  smflimsuplem8  41553
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