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Theorem climeqmpt 41854
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 5155 . 2 𝑥(𝑥𝐴𝐶)
3 nfmpt1 5155 . 2 𝑥(𝑥𝐵𝐶)
4 climeqmpt.m . 2 (𝜑𝑀 ∈ ℤ)
5 climeqmpt.z . 2 𝑍 = (ℤ𝑀)
6 climeqmpt.a . . 3 (𝜑𝐴𝑉)
76mptexd 6978 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ V)
8 climeqmpt.b . . 3 (𝜑𝐵𝑊)
98mptexd 6978 . 2 (𝜑 → (𝑥𝐵𝐶) ∈ V)
10 climeqmpt.s . . . . . 6 (𝜑𝑍𝐴)
1110adantr 481 . . . . 5 ((𝜑𝑥𝑍) → 𝑍𝐴)
12 simpr 485 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝑍)
1311, 12sseldd 3965 . . . 4 ((𝜑𝑥𝑍) → 𝑥𝐴)
14 climeqmpt.c . . . 4 ((𝜑𝑥𝑍) → 𝐶𝑈)
15 eqid 2818 . . . . 5 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1615fvmpt2 6771 . . . 4 ((𝑥𝐴𝐶𝑈) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1713, 14, 16syl2anc 584 . . 3 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
18 climeqmpt.t . . . . . . 7 (𝜑𝑍𝐵)
1918adantr 481 . . . . . 6 ((𝜑𝑥𝑍) → 𝑍𝐵)
2019, 12sseldd 3965 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝐵)
21 eqid 2818 . . . . . 6 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
2221fvmpt2 6771 . . . . 5 ((𝑥𝐵𝐶𝑈) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2320, 14, 22syl2anc 584 . . . 4 ((𝜑𝑥𝑍) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2423eqcomd 2824 . . 3 ((𝜑𝑥𝑍) → 𝐶 = ((𝑥𝐵𝐶)‘𝑥))
2517, 24eqtrd 2853 . 2 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐵𝐶)‘𝑥))
261, 2, 3, 4, 5, 7, 9, 25climeqf 41845 1 (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  Vcvv 3492  wss 3933   class class class wbr 5057  cmpt 5137  cfv 6348  cz 11969  cuz 12231  cli 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-z 11970  df-uz 12232  df-clim 14833
This theorem is referenced by:  smflimsuplem6  42976  smflimsuplem8  42978
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