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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 23124 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 23104 | . . . . . 6 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐹:𝐶⟶𝐷) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) |
4 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐷) |
5 | cncfcompt.bcn | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) | |
6 | cncff 23104 | . . . . . 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 6647 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
9 | 4, 8 | ffvelrnd 6624 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ 𝐷) |
10 | 9 | fmpttd 6649 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷) |
11 | cncfrss2 23103 | . . . 4 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐷 ⊆ ℂ) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
13 | eqidd 2779 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
14 | 3 | feqmptd 6509 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
15 | fveq2 6446 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
16 | 8, 13, 14, 15 | fmptco 6661 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
17 | ssid 3842 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
18 | cncfss 23110 | . . . . . . 7 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐶–cn→𝐷) ⊆ (𝐶–cn→ℂ)) | |
19 | 12, 17, 18 | sylancl 580 | . . . . . 6 ⊢ (𝜑 → (𝐶–cn→𝐷) ⊆ (𝐶–cn→ℂ)) |
20 | 19, 1 | sseldd 3822 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) |
21 | 5, 20 | cncfco 23118 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (𝐴–cn→ℂ)) |
22 | 16, 21 | eqeltrrd 2860 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→ℂ)) |
23 | cncffvrn 23109 | . . 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 579 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷) ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷)) |
25 | 10, 24 | mpbird 249 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 ⊆ wss 3792 ↦ cmpt 4965 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 –cn→ccncf 23087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 df-cj 14246 df-re 14247 df-im 14248 df-abs 14383 df-cncf 23089 |
This theorem is referenced by: itgsbtaddcnst 41125 fourierdlem23 41274 fourierdlem83 41333 fourierdlem101 41351 |
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