![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cncfmpt1f | Structured version Visualization version GIF version |
Description: Composition of continuous functions. –cn→ analogue of cnmpt11f 23137. (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 24378 | . . . . 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 2733 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
5 | 4 | fmpt 7097 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
6 | 3, 5 | sylibr 233 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ) |
7 | eqidd 2734 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
8 | cncfmpt1f.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) | |
9 | cncff 24378 | . . . . 5 ⊢ (𝐹 ∈ (ℂ–cn→ℂ) → 𝐹:ℂ⟶ℂ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
11 | 10 | feqmptd 6949 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦))) |
12 | fveq2 6881 | . . 3 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
13 | 6, 7, 11, 12 | fmptcof 7115 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴))) |
14 | 1, 8 | cncfco 24392 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝑋–cn→ℂ)) |
15 | 13, 14 | eqeltrrd 2835 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3062 ↦ cmpt 5227 ∘ ccom 5676 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 –cn→ccncf 24361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 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-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-2 12262 df-cj 15033 df-re 15034 df-im 15035 df-abs 15170 df-cncf 24363 |
This theorem is referenced by: taylthlem2 25855 sincn 25925 coscn 25926 pige3ALT 25998 efmul2picn 33539 itgexpif 33549 ftc1cnnclem 36464 ftc2nc 36475 itgcoscmulx 44558 itgsincmulx 44563 dirkeritg 44691 dirkercncflem2 44693 dirkercncflem4 44695 fourierdlem16 44712 fourierdlem21 44717 fourierdlem22 44718 fourierdlem39 44735 fourierdlem58 44753 fourierdlem62 44757 fourierdlem68 44763 fourierdlem73 44768 fourierdlem76 44771 fourierdlem78 44773 fourierdlem83 44778 sqwvfoura 44817 sqwvfourb 44818 etransclem18 44841 etransclem46 44869 |
Copyright terms: Public domain | W3C validator |