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Mirrors > Home > MPE Home > Th. List > cnmpt1k | Structured version Visualization version GIF version |
Description: The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptk1.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmptk1.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmptk1.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
cnmpt1k.m | ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) |
cnmpt1k.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) |
cnmpt1k.b | ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿))) |
cnmpt1k.c | ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cnmpt1k | ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmptk1.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | cnmptk1.l | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) | |
3 | cnmpt1k.a | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) | |
4 | cnf2 23273 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) |
6 | eqid 2735 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
7 | 6 | fmpt 7130 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) |
8 | 5, 7 | sylibr 234 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍) |
10 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
11 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)) | |
12 | cnmpt1k.c | . . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) | |
13 | 9, 10, 11, 12 | fmptcof 7150 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) |
14 | 13 | mpteq2dva 5248 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶))) |
15 | cnmptk1.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
16 | cnmpt1k.b | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿))) | |
17 | topontop 22935 | . . . . 5 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top) | |
18 | 2, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ Top) |
19 | cnmpt1k.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) | |
20 | topontop 22935 | . . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | eqid 2735 | . . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿) | |
23 | 22 | xkotopon 23624 | . . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀))) |
24 | 18, 21, 23 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀))) |
25 | 21, 3 | xkoco1cn 23681 | . . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ ((𝑀 ↑ko 𝐿) Cn (𝑀 ↑ko 𝐽))) |
26 | coeq1 5871 | . . 3 ⊢ (𝑤 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) | |
27 | 15, 16, 24, 25, 26 | cnmpt11 23687 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) |
28 | 14, 27 | eqeltrrd 2840 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) |
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: (None) |
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