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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 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
4 | cncfcompt2.ab | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) | |
5 | cncff 23498 | . . . . . . . . 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 | fvmptelrn 6854 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵) |
8 | 3, 7 | sseldd 3916 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐶) |
9 | 8 | ex 416 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑅 ∈ 𝐶)) |
10 | 1, 9 | ralrimi 3180 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐶) |
11 | eqidd 2799 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)) | |
12 | eqidd 2799 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) = (𝑦 ∈ 𝐶 ↦ 𝑆)) | |
13 | cncfcompt2.st | . . . 4 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) | |
14 | 10, 11, 12, 13 | fmptcof 6869 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
15 | 14 | eqcomd 2804 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) = ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅))) |
16 | cncfcompt2.cd | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) | |
17 | cncfrss 23496 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸) → 𝐶 ⊆ ℂ) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
19 | cncfss 23504 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) | |
20 | 2, 18, 19 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
21 | 20, 4 | sseldd 3916 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐶)) |
22 | 21, 16 | cncfco 23512 | . 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) ∈ (𝐴–cn→𝐸)) |
23 | 15, 22 | eqeltrd 2890 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ⊆ wss 3881 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 (class class class)co 7135 ℂcc 10524 –cn→ccncf 23481 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-cj 14450 df-re 14451 df-im 14452 df-abs 14587 df-cncf 23483 |
This theorem is referenced by: lcmineqlem9 39325 lcmineqlem12 39328 etransclem18 42894 etransclem22 42898 etransclem46 42922 |
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