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Mirrors > Home > MPE Home > Th. List > cnmpt22f | Structured version Visualization version GIF version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmpt21.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
cnmpt2t.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
cnmpt22f.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) |
Ref | Expression |
---|---|
cnmpt22f | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt21.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | cnmpt21.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
3 | cnmpt21.a | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | |
4 | cnmpt2t.b | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) | |
5 | cntop2 23232 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top) | |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Top) |
7 | toptopon2 22907 | . . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | |
8 | 6, 7 | sylib 217 | . 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
9 | cntop2 23232 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top) | |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Top) |
11 | toptopon2 22907 | . . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) | |
12 | 10, 11 | sylib 217 | . 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
13 | txtopon 23582 | . . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀))) | |
14 | 8, 12, 13 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀))) |
15 | cnmpt22f.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) | |
16 | cntop2 23232 | . . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top) | |
17 | 15, 16 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top) |
18 | toptopon2 22907 | . . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) | |
19 | 17, 18 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
20 | cnf2 23240 | . . . . . 6 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁) | |
21 | 14, 19, 15, 20 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁) |
22 | 21 | ffnd 6720 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀)) |
23 | fnov 7548 | . . . 4 ⊢ (𝐹 Fn (∪ 𝐿 × ∪ 𝑀) ↔ 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤))) | |
24 | 22, 23 | sylib 217 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤))) |
25 | 24, 15 | eqeltrrd 2827 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) |
26 | oveq12 7424 | . 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵)) | |
27 | 1, 2, 3, 4, 8, 12, 25, 26 | cnmpt22 23665 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4907 × cxp 5672 Fn wfn 6540 ⟶wf 6541 ‘cfv 6545 (class class class)co 7415 ∈ cmpo 7417 Topctop 22882 TopOnctopon 22899 Cn ccn 23215 ×t ctx 23551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-fv 6553 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7994 df-2nd 7995 df-map 8848 df-topgen 17452 df-top 22883 df-topon 22900 df-bases 22936 df-cn 23218 df-tx 23553 |
This theorem is referenced by: cnmptcom 23669 cnmpt2plusg 24079 istgp2 24082 cnmpt2vsca 24186 cnmpt2ds 24846 divcnOLD 24871 cnrehmeo 24965 cnrehmeoOLD 24966 htpycom 24989 htpyco1 24991 htpycc 24993 reparphti 25010 reparphtiOLD 25011 pcohtpylem 25033 cnmpt2ip 25263 cxpcnOLD 26769 vmcn 30628 dipcn 30649 mndpluscn 33753 cvxsconn 35083 |
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