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Mirrors > Home > MPE Home > Th. List > cncfcompt2 | Structured version Visualization version GIF version |
Description: Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
cncfcompt2.xph | ⊢ Ⅎ𝑥𝜑 |
cncfcompt2.ab | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) |
cncfcompt2.cd | ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) |
cncfcompt2.bc | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
cncfcompt2.st | ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) |
Ref | Expression |
---|---|
cncfcompt2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfcompt2.xph | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
2 | cncfcompt2.bc | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | 2 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
4 | cncfcompt2.ab | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) | |
5 | cncff 24279 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵) |
7 | 6 | fvmptelcdm 7065 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵) |
8 | 3, 7 | sseldd 3949 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐶) |
9 | 8 | ex 414 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑅 ∈ 𝐶)) |
10 | 1, 9 | ralrimi 3239 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐶) |
11 | eqidd 2734 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)) | |
12 | eqidd 2734 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) = (𝑦 ∈ 𝐶 ↦ 𝑆)) | |
13 | cncfcompt2.st | . . . 4 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) | |
14 | 10, 11, 12, 13 | fmptcof 7080 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
15 | 14 | eqcomd 2739 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) = ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅))) |
16 | cncfcompt2.cd | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) | |
17 | cncfrss 24277 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸) → 𝐶 ⊆ ℂ) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
19 | cncfss 24285 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) | |
20 | 2, 18, 19 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
21 | 20, 4 | sseldd 3949 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐶)) |
22 | 21, 16 | cncfco 24293 | . 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) ∈ (𝐴–cn→𝐸)) |
23 | 15, 22 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ⊆ wss 3914 ↦ cmpt 5192 ∘ ccom 5641 ⟶wf 6496 (class class class)co 7361 ℂcc 11057 –cn→ccncf 24262 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-2 12224 df-cj 14993 df-re 14994 df-im 14995 df-abs 15130 df-cncf 24264 |
This theorem is referenced by: lcmineqlem9 40544 lcmineqlem12 40547 etransclem18 44583 etransclem22 44587 etransclem46 44611 |
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