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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 22272. (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 23501 | . . . . 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 2821 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
5 | 4 | fmpt 6874 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
6 | 3, 5 | sylibr 236 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ) |
7 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
8 | cncfmpt1f.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | |
9 | cncff 23501 | . . . . 5 ⊢ (𝐹 ∈ (ℂ–cn→ℂ) → 𝐹:ℂ⟶ℂ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
11 | 10 | feqmptd 6733 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦))) |
12 | fveq2 6670 | . . 3 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
13 | 6, 7, 11, 12 | fmptcof 6892 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴))) |
14 | 1, 8 | cncfco 23515 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝑋–cn→ℂ)) |
15 | 13, 14 | eqeltrrd 2914 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3138 ↦ cmpt 5146 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 –cn→ccncf 23484 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 df-abs 14595 df-cncf 23486 |
This theorem is referenced by: taylthlem2 24962 sincn 25032 coscn 25033 pige3ALT 25105 efmul2picn 31867 itgexpif 31877 ftc1cnnclem 34980 ftc2nc 34991 itgcoscmulx 42274 itgsincmulx 42279 dirkeritg 42407 dirkercncflem2 42409 dirkercncflem4 42411 fourierdlem16 42428 fourierdlem21 42433 fourierdlem22 42434 fourierdlem39 42451 fourierdlem58 42469 fourierdlem62 42473 fourierdlem68 42479 fourierdlem73 42484 fourierdlem76 42487 fourierdlem78 42489 fourierdlem83 42494 sqwvfoura 42533 sqwvfourb 42534 etransclem18 42557 etransclem46 42585 |
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