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Theorem climeqmpt 42205
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 5151 . 2 𝑥(𝑥𝐴𝐶)
3 nfmpt1 5151 . 2 𝑥(𝑥𝐵𝐶)
4 climeqmpt.m . 2 (𝜑𝑀 ∈ ℤ)
5 climeqmpt.z . 2 𝑍 = (ℤ𝑀)
6 climeqmpt.a . . 3 (𝜑𝐴𝑉)
76mptexd 6976 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ V)
8 climeqmpt.b . . 3 (𝜑𝐵𝑊)
98mptexd 6976 . 2 (𝜑 → (𝑥𝐵𝐶) ∈ V)
10 climeqmpt.s . . . . . 6 (𝜑𝑍𝐴)
1110adantr 484 . . . . 5 ((𝜑𝑥𝑍) → 𝑍𝐴)
12 simpr 488 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝑍)
1311, 12sseldd 3954 . . . 4 ((𝜑𝑥𝑍) → 𝑥𝐴)
14 climeqmpt.c . . . 4 ((𝜑𝑥𝑍) → 𝐶𝑈)
15 eqid 2824 . . . . 5 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1615fvmpt2 6768 . . . 4 ((𝑥𝐴𝐶𝑈) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1713, 14, 16syl2anc 587 . . 3 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
18 climeqmpt.t . . . . . . 7 (𝜑𝑍𝐵)
1918adantr 484 . . . . . 6 ((𝜑𝑥𝑍) → 𝑍𝐵)
2019, 12sseldd 3954 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝐵)
21 eqid 2824 . . . . . 6 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
2221fvmpt2 6768 . . . . 5 ((𝑥𝐵𝐶𝑈) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2320, 14, 22syl2anc 587 . . . 4 ((𝜑𝑥𝑍) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2423eqcomd 2830 . . 3 ((𝜑𝑥𝑍) → 𝐶 = ((𝑥𝐵𝐶)‘𝑥))
2517, 24eqtrd 2859 . 2 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐵𝐶)‘𝑥))
261, 2, 3, 4, 5, 7, 9, 25climeqf 42196 1 (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2115  Vcvv 3480  wss 3919   class class class wbr 5053  cmpt 5133  cfv 6344  cz 11976  cuz 12238  cli 14839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-neg 10867  df-z 11977  df-uz 12239  df-clim 14843
This theorem is referenced by:  smflimsuplem6  43322  smflimsuplem8  43324
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