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Mirrors > Home > ILE Home > Th. List > cnco | GIF version |
Description: The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnco | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 12370 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
2 | cntop2 12371 | . . 3 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top) | |
3 | 1, 2 | anim12i 336 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top)) |
4 | eqid 2139 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | eqid 2139 | . . . . 5 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
6 | 4, 5 | cnf 12373 | . . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐺:∪ 𝐾⟶∪ 𝐿) |
7 | eqid 2139 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7, 4 | cnf 12373 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
9 | fco 5288 | . . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿) | |
10 | 6, 8, 9 | syl2anr 288 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿) |
11 | cnvco 4724 | . . . . . . 7 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
12 | 11 | imaeq1i 4878 | . . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥) |
13 | imaco 5044 | . . . . . 6 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥)) | |
14 | 12, 13 | eqtri 2160 | . . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥)) |
15 | simpll 518 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
16 | cnima 12389 | . . . . . . 7 ⊢ ((𝐺 ∈ (𝐾 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾) | |
17 | 16 | adantll 467 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾) |
18 | cnima 12389 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (◡𝐺 “ 𝑥) ∈ 𝐾) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽) | |
19 | 15, 17, 18 | syl2anc 408 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽) |
20 | 14, 19 | eqeltrid 2226 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽) |
21 | 20 | ralrimiva 2505 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽) |
22 | 10, 21 | jca 304 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)) |
23 | 7, 5 | iscn2 12369 | . 2 ⊢ ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿) ↔ ((𝐽 ∈ Top ∧ 𝐿 ∈ Top) ∧ ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽))) |
24 | 3, 22, 23 | sylanbrc 413 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∀wral 2416 ∪ cuni 3736 ◡ccnv 4538 “ cima 4542 ∘ ccom 4543 ⟶wf 5119 (class class class)co 5774 Topctop 12164 Cn ccn 12354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12165 df-topon 12178 df-cn 12357 |
This theorem is referenced by: txcn 12444 cnmpt11 12452 cnmpt21 12460 hmeoco 12485 |
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