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Mirrors > Home > MPE Home > Th. List > cncfmpt1f | Structured version Visualization version GIF version |
Description: Composition of continuous functions. –cn→ analogue of cnmpt11f 22274. (Contributed by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
cncfmpt1f.1 | ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
cncfmpt1f.2 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) |
Ref | Expression |
---|---|
cncfmpt1f | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt1f.2 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) | |
2 | cncff 23503 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
4 | eqid 2823 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
5 | 4 | fmpt 6876 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
6 | 3, 5 | sylibr 236 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ) |
7 | eqidd 2824 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
8 | cncfmpt1f.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | |
9 | cncff 23503 | . . . . 5 ⊢ (𝐹 ∈ (ℂ–cn→ℂ) → 𝐹:ℂ⟶ℂ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
11 | 10 | feqmptd 6735 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦))) |
12 | fveq2 6672 | . . 3 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
13 | 6, 7, 11, 12 | fmptcof 6894 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴))) |
14 | 1, 8 | cncfco 23517 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝑋–cn→ℂ)) |
15 | 13, 14 | eqeltrrd 2916 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3140 ↦ cmpt 5148 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 –cn→ccncf 23486 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-abs 14597 df-cncf 23488 |
This theorem is referenced by: taylthlem2 24964 sincn 25034 coscn 25035 pige3ALT 25107 efmul2picn 31869 itgexpif 31879 ftc1cnnclem 34967 ftc2nc 34978 itgcoscmulx 42261 itgsincmulx 42266 dirkeritg 42394 dirkercncflem2 42396 dirkercncflem4 42398 fourierdlem16 42415 fourierdlem21 42420 fourierdlem22 42421 fourierdlem39 42438 fourierdlem58 42456 fourierdlem62 42460 fourierdlem68 42466 fourierdlem73 42471 fourierdlem76 42474 fourierdlem78 42476 fourierdlem83 42481 sqwvfoura 42520 sqwvfourb 42521 etransclem18 42544 etransclem46 42572 |
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