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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 42887 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 7040 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) |
4 | climfveqmpt3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | 4 | mptexd 7040 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V) |
6 | climfveqmpt3.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climfveqmpt3.k | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
8 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
9 | 7, 8 | nfan 1907 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑘𝑗 | |
11 | 10 | nfcsb1 3835 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 |
12 | 10 | nfcsb1 3835 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷 |
13 | 11, 12 | nfeq 2917 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷 |
14 | 9, 13 | nfim 1904 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
15 | eleq1w 2820 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 632 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | csbeq1a 3825 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
18 | csbeq1a 3825 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷) | |
19 | 17, 18 | eqeq12d 2753 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)) |
20 | 16, 19 | imbi12d 348 | . . . 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 2238 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
23 | climfveqmpt3.i | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
24 | 23 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
25 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
26 | 24, 25 | sseldd 3902 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴) |
27 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑘𝑈 | |
28 | 11, 27 | nfel 2918 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈 |
29 | 9, 28 | nfim 1904 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
30 | 17 | eleq1d 2822 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)) |
31 | 16, 30 | imbi12d 348 | . . . . 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 2238 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
34 | eqid 2737 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
35 | 10, 11, 17, 34 | fvmptf 6839 | . . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
36 | 26, 33, 35 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
37 | climfveqmpt3.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐶) | |
38 | 37 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐶) |
39 | 38, 25 | sseldd 3902 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶) |
40 | 22, 33 | eqeltrrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) |
41 | eqid 2737 | . . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷) | |
42 | 10, 12, 18, 41 | fvmptf 6839 | . . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
43 | 39, 40, 42 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
44 | 22, 36, 43 | 3eqtr4d 2787 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗)) |
45 | 1, 3, 5, 6, 44 | climfveq 42885 | 1 ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Vcvv 3408 ⦋csb 3811 ⊆ wss 3866 ↦ cmpt 5135 ‘cfv 6380 ℤcz 12176 ℤ≥cuz 12438 ⇝ cli 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 |
This theorem is referenced by: smflimmpt 44015 |
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