Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfcompt | Structured version Visualization version GIF version |
Description: Composition of continuous functions. A generalization of cncfmpt1f 23448 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cncfcompt.bcn | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) |
cncfcompt.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→𝐷)) |
Ref | Expression |
---|---|
cncfcompt | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfcompt.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→𝐷)) | |
2 | cncff 23428 | . . . . . 6 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐹:𝐶⟶𝐷) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐷) |
5 | cncfcompt.bcn | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) | |
6 | cncff 23428 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
8 | 7 | fvmptelrn 6869 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
9 | 4, 8 | ffvelrnd 6844 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ 𝐷) |
10 | 9 | fmpttd 6871 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷) |
11 | cncfrss2 23427 | . . . 4 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐷 ⊆ ℂ) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
13 | eqidd 2819 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
14 | 3 | feqmptd 6726 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
15 | fveq2 6663 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
16 | 8, 13, 14, 15 | fmptco 6883 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
17 | ssid 3986 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
18 | cncfss 23434 | . . . . . . 7 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐶–cn→𝐷) ⊆ (𝐶–cn→ℂ)) | |
19 | 12, 17, 18 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → (𝐶–cn→𝐷) ⊆ (𝐶–cn→ℂ)) |
20 | 19, 1 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) |
21 | 5, 20 | cncfco 23442 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (𝐴–cn→ℂ)) |
22 | 16, 21 | eqeltrrd 2911 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→ℂ)) |
23 | cncffvrn 23433 | . . 3 ⊢ ((𝐷 ⊆ ℂ ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→ℂ)) → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷) ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷)) | |
24 | 12, 22, 23 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷) ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷)) |
25 | 10, 24 | mpbird 258 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-cj 14446 df-re 14447 df-im 14448 df-abs 14583 df-cncf 23413 |
This theorem is referenced by: itgsbtaddcnst 42143 fourierdlem23 42292 fourierdlem83 42351 fourierdlem101 42369 |
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