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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeldmeqf | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climeldmeqf.p | ⊢ Ⅎ𝑘𝜑 |
climeldmeqf.n | ⊢ Ⅎ𝑘𝐹 |
climeldmeqf.o | ⊢ Ⅎ𝑘𝐺 |
climeldmeqf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeldmeqf.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climeldmeqf.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climeldmeqf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeldmeqf.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climeldmeqf | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeldmeqf.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climeldmeqf.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | climeldmeqf.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | climeldmeqf.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climeldmeqf.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
6 | nfv 1910 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1895 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climeldmeqf.n | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
9 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6907 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
11 | climeldmeqf.o | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
12 | 11, 9 | nffv 6907 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
13 | 10, 12 | nfeq 2913 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
14 | 7, 13 | nfim 1892 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
15 | eleq1w 2812 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 629 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fveq2 6897 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
18 | fveq2 6897 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
19 | 17, 18 | eqeq12d 2744 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
20 | 16, 19 | imbi12d 344 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)))) |
21 | climeldmeqf.e | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
22 | 14, 20, 21 | chvarfv 2229 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
23 | 1, 2, 3, 4, 22 | climeldmeq 45053 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 dom cdm 5678 ‘cfv 6548 ℤcz 12589 ℤ≥cuz 12853 ⇝ cli 15461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 |
This theorem is referenced by: climeldmeqmpt2 45083 |
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