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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 21785 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) | |
5 | 1, 2, 3, 4 | syl3anc 1363 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) |
6 | eqid 2818 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
7 | 6 | fmpt 6866 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍) |
8 | 5, 7 | sylibr 235 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍) |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍) |
10 | eqidd 2819 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
11 | eqidd 2819 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)) | |
12 | cnmpt1k.c | . . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) | |
13 | 9, 10, 11, 12 | fmptcof 6884 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) |
14 | 13 | mpteq2dva 5152 | . 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 21449 | . . . . 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 21449 | . . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Top) |
22 | eqid 2818 | . . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿) | |
23 | 22 | xkotopon 22136 | . . . 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 22193 | . . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ ((𝑀 ↑ko 𝐿) Cn (𝑀 ↑ko 𝐽))) |
26 | coeq1 5721 | . . 3 ⊢ (𝑤 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) | |
27 | 15, 16, 24, 25, 26 | cnmpt11 22199 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) |
28 | 14, 27 | eqeltrrd 2911 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 ↑ko cxko 22097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-fin 8501 df-fi 8863 df-rest 16684 df-topgen 16705 df-top 21430 df-topon 21447 df-bases 21482 df-cn 21763 df-cmp 21923 df-xko 22099 |
This theorem is referenced by: (None) |
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