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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 24940 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 24919 | . . . . . 6 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐹:𝐶⟶𝐷) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) |
| 4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐷) |
| 5 | cncfcompt.bcn | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) | |
| 6 | cncff 24919 | . . . . . 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 7133 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
| 9 | 4, 8 | ffvelcdmd 7105 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ 𝐷) |
| 10 | 9 | fmpttd 7135 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶𝐷) |
| 11 | cncfrss2 24918 | . . . 4 ⊢ (𝐹 ∈ (𝐶–cn→𝐷) → 𝐷 ⊆ ℂ) | |
| 12 | 1, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
| 13 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
| 14 | 3 | feqmptd 6977 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
| 15 | fveq2 6906 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
| 16 | 8, 13, 14, 15 | fmptco 7149 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
| 17 | ssid 4006 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
| 18 | cncfss 24925 | . . . . . . 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 3984 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) |
| 21 | 5, 20 | cncfco 24933 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (𝐴–cn→ℂ)) |
| 22 | 16, 21 | eqeltrrd 2842 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→ℂ)) |
| 23 | cncfcdm 24924 | . . 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 257 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ (𝐴–cn→𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 ↦ cmpt 5225 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 df-abs 15275 df-cncf 24904 |
| This theorem is referenced by: itgsbtaddcnst 45997 fourierdlem23 46145 fourierdlem83 46204 fourierdlem101 46222 |
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