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Theorem climeqmpt 40430
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 4897 . 2 𝑥(𝑥𝐴𝐶)
3 nfmpt1 4897 . 2 𝑥(𝑥𝐵𝐶)
4 climeqmpt.m . 2 (𝜑𝑀 ∈ ℤ)
5 climeqmpt.z . 2 𝑍 = (ℤ𝑀)
6 climeqmpt.a . . 3 (𝜑𝐴𝑉)
76mptexd 6649 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ V)
8 climeqmpt.b . . 3 (𝜑𝐵𝑊)
98mptexd 6649 . 2 (𝜑 → (𝑥𝐵𝐶) ∈ V)
10 climeqmpt.s . . . . . 6 (𝜑𝑍𝐴)
1110adantr 472 . . . . 5 ((𝜑𝑥𝑍) → 𝑍𝐴)
12 simpr 479 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝑍)
1311, 12sseldd 3743 . . . 4 ((𝜑𝑥𝑍) → 𝑥𝐴)
14 climeqmpt.c . . . 4 ((𝜑𝑥𝑍) → 𝐶𝑈)
15 eqid 2758 . . . . 5 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1615fvmpt2 6451 . . . 4 ((𝑥𝐴𝐶𝑈) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1713, 14, 16syl2anc 696 . . 3 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
18 climeqmpt.t . . . . . . 7 (𝜑𝑍𝐵)
1918adantr 472 . . . . . 6 ((𝜑𝑥𝑍) → 𝑍𝐵)
2019, 12sseldd 3743 . . . . 5 ((𝜑𝑥𝑍) → 𝑥𝐵)
21 eqid 2758 . . . . . 6 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
2221fvmpt2 6451 . . . . 5 ((𝑥𝐵𝐶𝑈) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2320, 14, 22syl2anc 696 . . . 4 ((𝜑𝑥𝑍) → ((𝑥𝐵𝐶)‘𝑥) = 𝐶)
2423eqcomd 2764 . . 3 ((𝜑𝑥𝑍) → 𝐶 = ((𝑥𝐵𝐶)‘𝑥))
2517, 24eqtrd 2792 . 2 ((𝜑𝑥𝑍) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐵𝐶)‘𝑥))
261, 2, 3, 4, 5, 7, 9, 25climeqf 40421 1 (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wnf 1855  wcel 2137  Vcvv 3338  wss 3713   class class class wbr 4802  cmpt 4879  cfv 6047  cz 11567  cuz 11877  cli 14412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-pre-lttri 10200  ax-pre-lttrn 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-po 5185  df-so 5186  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-neg 10459  df-z 11568  df-uz 11878  df-clim 14416
This theorem is referenced by:  smflimsuplem6  41535  smflimsuplem8  41537
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