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Theorem cnmpt1k 21814
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.)
Hypotheses
Ref Expression
cnmptk1.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt1k.m (𝜑𝑀 ∈ (TopOn‘𝑊))
cnmpt1k.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))
cnmpt1k.b (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀 ^ko 𝐿)))
cnmpt1k.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmpt1k (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑧,𝑍,𝑦   𝑧,𝐴   𝑥,𝐵   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑧,𝐶   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥)   𝐵(𝑦,𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)   𝑀(𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem cnmpt1k
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cnmptk1.j . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmptk1.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
3 cnmpt1k.a . . . . . . 7 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))
4 cnf2 21382 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐴):𝑋𝑍)
51, 2, 3, 4syl3anc 1491 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑍)
6 eqid 2799 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6606 . . . . . 6 (∀𝑥𝑋 𝐴𝑍 ↔ (𝑥𝑋𝐴):𝑋𝑍)
85, 7sylibr 226 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑍)
98adantr 473 . . . 4 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴𝑍)
10 eqidd 2800 . . . 4 ((𝜑𝑦𝑌) → (𝑥𝑋𝐴) = (𝑥𝑋𝐴))
11 eqidd 2800 . . . 4 ((𝜑𝑦𝑌) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
12 cnmpt1k.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
139, 10, 11, 12fmptcof 6624 . . 3 ((𝜑𝑦𝑌) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
1413mpteq2dva 4937 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) = (𝑦𝑌 ↦ (𝑥𝑋𝐶)))
15 cnmptk1.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
16 cnmpt1k.b . . 3 (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀 ^ko 𝐿)))
17 topontop 21046 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
182, 17syl 17 . . . 4 (𝜑𝐿 ∈ Top)
19 cnmpt1k.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
20 topontop 21046 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
2119, 20syl 17 . . . 4 (𝜑𝑀 ∈ Top)
22 eqid 2799 . . . . 5 (𝑀 ^ko 𝐿) = (𝑀 ^ko 𝐿)
2322xkotopon 21732 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2418, 21, 23syl2anc 580 . . 3 (𝜑 → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2521, 3xkoco1cn 21789 . . 3 (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥𝑋𝐴))) ∈ ((𝑀 ^ko 𝐿) Cn (𝑀 ^ko 𝐽)))
26 coeq1 5483 . . 3 (𝑤 = (𝑧𝑍𝐵) → (𝑤 ∘ (𝑥𝑋𝐴)) = ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)))
2715, 16, 24, 25, 26cnmpt11 21795 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
2814, 27eqeltrrd 2879 1 (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  cmpt 4922  ccom 5316  wf 6097  cfv 6101  (class class class)co 6878  Topctop 21026  TopOnctopon 21043   Cn ccn 21357   ^ko cxko 21693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-fin 8199  df-fi 8559  df-rest 16398  df-topgen 16419  df-top 21027  df-topon 21044  df-bases 21079  df-cn 21360  df-cmp 21519  df-xko 21695
This theorem is referenced by: (None)
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