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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  Ⅎwnf 1785   ∈ wcel 2115  Vcvv 3480   ⊆ wss 3919   class class class wbr 5053   ↦ cmpt 5133  ‘cfv 6344  ℤcz 11976  ℤ≥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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