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Mirrors > Home > MPE Home > Th. List > cnmptk1 | Structured version Visualization version GIF version |
Description: The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) |
Ref | Expression |
---|---|
cnmptk1.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmptk1.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmptk1.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
cnmptk1.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) |
cnmptk1.b | ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) |
cnmptk1.c | ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cnmptk1 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmptk1.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
3 | cnmptk1.l | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍)) |
5 | cnmptk1.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
6 | topontop 22935 | . . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) | |
7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top) |
8 | topontop 22935 | . . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top) | |
9 | 3, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top) |
10 | eqid 2735 | . . . . . . . . . 10 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾) | |
11 | 10 | xkotopon 23624 | . . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿))) |
12 | 7, 9, 11 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿))) |
13 | cnmptk1.a | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | |
14 | cnf2 23273 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿)) | |
15 | 5, 12, 13, 14 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿)) |
16 | 15 | fvmptelcdm 7133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
17 | cnf2 23273 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍) | |
18 | 2, 4, 16, 17 | syl3anc 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍) |
19 | eqid 2735 | . . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴) | |
20 | 19 | fmpt 7130 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍) |
21 | 18, 20 | sylibr 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍) |
22 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)) | |
23 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)) | |
24 | cnmptk1.c | . . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) | |
25 | 21, 22, 23, 24 | fmptcof 7150 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑦 ∈ 𝑌 ↦ 𝐶)) |
26 | 25 | mpteq2dva 5248 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) |
27 | cnmptk1.b | . . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) | |
28 | 7, 27 | xkoco2cn 23682 | . . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑤)) ∈ ((𝐿 ↑ko 𝐾) Cn (𝑀 ↑ko 𝐾))) |
29 | coeq2 5872 | . . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑤) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) | |
30 | 5, 13, 12, 28, 29 | cnmpt11 23687 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾))) |
31 | 26, 30 | eqeltrrd 2840 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Topctop 22915 TopOnctopon 22932 Cn ccn 23248 ↑ko cxko 23585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-map 8867 df-en 8985 df-dom 8986 df-fin 8988 df-fi 9449 df-rest 17469 df-topgen 17490 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 df-cmp 23411 df-xko 23587 |
This theorem is referenced by: cnmpt2k 23712 |
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