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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 14178 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
2 | cntop2 14179 | . . 3 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top) | |
3 | 1, 2 | anim12i 338 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top)) |
4 | eqid 2189 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | eqid 2189 | . . . . 5 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
6 | 4, 5 | cnf 14181 | . . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐺:∪ 𝐾⟶∪ 𝐿) |
7 | eqid 2189 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7, 4 | cnf 14181 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
9 | fco 5400 | . . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿) | |
10 | 6, 8, 9 | syl2anr 290 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿) |
11 | cnvco 4830 | . . . . . . 7 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
12 | 11 | imaeq1i 4985 | . . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥) |
13 | imaco 5152 | . . . . . 6 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥)) | |
14 | 12, 13 | eqtri 2210 | . . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥)) |
15 | simpll 527 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
16 | cnima 14197 | . . . . . . 7 ⊢ ((𝐺 ∈ (𝐾 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾) | |
17 | 16 | adantll 476 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾) |
18 | cnima 14197 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (◡𝐺 “ 𝑥) ∈ 𝐾) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽) | |
19 | 15, 17, 18 | syl2anc 411 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽) |
20 | 14, 19 | eqeltrid 2276 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽) |
21 | 20 | ralrimiva 2563 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽) |
22 | 10, 21 | jca 306 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)) |
23 | 7, 5 | iscn2 14177 | . 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 417 | 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 104 ∈ wcel 2160 ∀wral 2468 ∪ cuni 3824 ◡ccnv 4643 “ cima 4647 ∘ ccom 4648 ⟶wf 5231 (class class class)co 5897 Topctop 13974 Cn ccn 14162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-map 6677 df-top 13975 df-topon 13988 df-cn 14165 |
This theorem is referenced by: txcn 14252 cnmpt11 14260 cnmpt21 14268 hmeoco 14293 |
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