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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeldmeqmpt2 | 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 |
---|---|
climeldmeqmpt2.k | ⊢ Ⅎ𝑘𝜑 |
climeldmeqmpt2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeldmeqmpt2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeldmeqmpt2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
climeldmeqmpt2.t | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
climeldmeqmpt2.i | ⊢ (𝜑 → 𝑍 ⊆ 𝐴) |
climeldmeqmpt2.l | ⊢ (𝜑 → 𝑍 ⊆ 𝐵) |
climeldmeqmpt2.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
climeldmeqmpt2 | ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeldmeqmpt2.k | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | nfmpt1 4982 | . 2 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 4982 | . 2 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐵 ↦ 𝐶) | |
4 | climeldmeqmpt2.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | climeldmeqmpt2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
6 | 5 | mptexd 6759 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) ∈ V) |
7 | climeldmeqmpt2.t | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
8 | 7 | mptexd 6759 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ V) |
9 | climeldmeqmpt2.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | climeldmeqmpt2.i | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | sselda 3821 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴) |
12 | climeldmeqmpt2.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
13 | fvmpt4 40361 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) | |
14 | 11, 12, 13 | syl2anc 579 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
15 | climeldmeqmpt2.l | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
16 | 15 | sselda 3821 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐵) |
17 | fvmpt4 40361 | . . . 4 ⊢ ((𝑘 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = 𝐶) | |
18 | 16, 12, 17 | syl2anc 579 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = 𝐶) |
19 | 14, 18 | eqtr4d 2817 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘)) |
20 | 1, 2, 3, 4, 6, 8, 9, 19 | climeldmeqf 40823 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 Ⅎwnf 1827 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 ↦ cmpt 4965 dom cdm 5355 ‘cfv 6135 ℤcz 11728 ℤ≥cuz 11992 ⇝ cli 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 |
This theorem is referenced by: (None) |
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