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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 5250 | . 2 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 5250 | . 2 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐵 ↦ 𝐶) | |
4 | climeldmeqmpt2.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | climeldmeqmpt2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
6 | 5 | mptexd 7230 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) ∈ V) |
7 | climeldmeqmpt2.t | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
8 | 7 | mptexd 7230 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ V) |
9 | climeldmeqmpt2.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | climeldmeqmpt2.i | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | sselda 3978 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴) |
12 | climeldmeqmpt2.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
13 | fvmpt4 44536 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) | |
14 | 11, 12, 13 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
15 | climeldmeqmpt2.l | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
16 | 15 | sselda 3978 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐵) |
17 | fvmpt4 44536 | . . . 4 ⊢ ((𝑘 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = 𝐶) | |
18 | 16, 12, 17 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = 𝐶) |
19 | 14, 18 | eqtr4d 2770 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘)) |
20 | 1, 2, 3, 4, 6, 8, 9, 19 | climeldmeqf 44994 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 ↦ cmpt 5225 dom cdm 5672 ‘cfv 6542 ℤcz 12580 ℤ≥cuz 12844 ⇝ cli 15452 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 |
This theorem is referenced by: (None) |
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