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Mirrors > Home > MPE Home > Th. List > Mathboxes > climfveqmpt3 | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 41972 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climfveqmpt3.k | ⊢ Ⅎ𝑘𝜑 |
climfveqmpt3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climfveqmpt3.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climfveqmpt3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
climfveqmpt3.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
climfveqmpt3.i | ⊢ (𝜑 → 𝑍 ⊆ 𝐴) |
climfveqmpt3.s | ⊢ (𝜑 → 𝑍 ⊆ 𝐶) |
climfveqmpt3.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) |
climfveqmpt3.d | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
climfveqmpt3 | ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climfveqmpt3.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climfveqmpt3.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2 | mptexd 6987 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) |
4 | climfveqmpt3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | 4 | mptexd 6987 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V) |
6 | climfveqmpt3.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climfveqmpt3.k | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
8 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
9 | 7, 8 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘𝑗 | |
11 | 10 | nfcsb1 3906 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 |
12 | 10 | nfcsb1 3906 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷 |
13 | 11, 12 | nfeq 2991 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷 |
14 | 9, 13 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
15 | eleq1w 2895 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | csbeq1a 3897 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
18 | csbeq1a 3897 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷) | |
19 | 17, 18 | eqeq12d 2837 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)) |
20 | 16, 19 | imbi12d 347 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷))) |
21 | climfveqmpt3.d | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) | |
22 | 14, 20, 21 | chvarfv 2242 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
23 | climfveqmpt3.i | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
24 | 23 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
25 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
26 | 24, 25 | sseldd 3968 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴) |
27 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘𝑈 | |
28 | 11, 27 | nfel 2992 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈 |
29 | 9, 28 | nfim 1897 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
30 | 17 | eleq1d 2897 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)) |
31 | 16, 30 | imbi12d 347 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈))) |
32 | climfveqmpt3.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) | |
33 | 29, 31, 32 | chvarfv 2242 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
34 | eqid 2821 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
35 | 10, 11, 17, 34 | fvmptf 6789 | . . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
36 | 26, 33, 35 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
37 | climfveqmpt3.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐶) | |
38 | 37 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐶) |
39 | 38, 25 | sseldd 3968 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶) |
40 | 22, 33 | eqeltrrd 2914 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) |
41 | eqid 2821 | . . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷) | |
42 | 10, 12, 18, 41 | fvmptf 6789 | . . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
43 | 39, 40, 42 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
44 | 22, 36, 43 | 3eqtr4d 2866 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗)) |
45 | 1, 3, 5, 6, 44 | climfveq 41970 | 1 ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3494 ⦋csb 3883 ⊆ wss 3936 ↦ cmpt 5146 ‘cfv 6355 ℤcz 11982 ℤ≥cuz 12244 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 |
This theorem is referenced by: smflimmpt 43104 |
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