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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 24429 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 24408 | . . . . . 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 24408 | . . . . . 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 | fvmptelcdm 7112 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
9 | 4, 8 | ffvelcdmd 7087 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ 𝐷) |
10 | 9 | fmpttd 7114 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷) |
11 | cncfrss2 24407 | . . . 4 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐷 ⊆ ℂ) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
13 | eqidd 2733 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
14 | 3 | feqmptd 6960 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
15 | fveq2 6891 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
16 | 8, 13, 14, 15 | fmptco 7126 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
17 | ssid 4004 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
18 | cncfss 24414 | . . . . . . 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 3983 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) |
21 | 5, 20 | cncfco 24422 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (𝐴–cn→ℂ)) |
22 | 16, 21 | eqeltrrd 2834 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→ℂ)) |
23 | cncfcdm 24413 | . . 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 256 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3948 ↦ cmpt 5231 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 –cn→ccncf 24391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-cj 15045 df-re 15046 df-im 15047 df-abs 15182 df-cncf 24393 |
This theorem is referenced by: itgsbtaddcnst 44688 fourierdlem23 44836 fourierdlem83 44895 fourierdlem101 44913 |
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