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Theorem cnmptk1 23540
Description: The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
cnmptk1.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptk1.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
cnmptk1.b (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
cnmptk1.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmptk1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
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,𝐵
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥,𝑦)   𝐵(𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)   𝑀(𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem cnmptk1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cnmptk1.k . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
21adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3 cnmptk1.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
43adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5 cnmptk1.j . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 22770 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
8 topontop 22770 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
93, 8syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
10 eqid 2726 . . . . . . . . . 10 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1110xkotopon 23459 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
127, 9, 11syl2anc 583 . . . . . . . 8 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
13 cnmptk1.a . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
14 cnf2 23108 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
155, 12, 13, 14syl3anc 1368 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
1615fvmptelcdm 7108 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
17 cnf2 23108 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
182, 4, 16, 17syl3anc 1368 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
19 eqid 2726 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2019fmpt 7105 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2118, 20sylibr 233 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
22 eqidd 2727 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
23 eqidd 2727 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
24 cnmptk1.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2521, 22, 23, 24fmptcof 7124 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
2625mpteq2dva 5241 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
27 cnmptk1.b . . . 4 (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
287, 27xkoco2cn 23517 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ ((𝑧𝑍𝐵) ∘ 𝑤)) ∈ ((𝐿ko 𝐾) Cn (𝑀ko 𝐾)))
29 coeq2 5852 . . 3 (𝑤 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑤) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
305, 13, 12, 28, 29cnmpt11 23522 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀ko 𝐾)))
3126, 30eqeltrrd 2828 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  cmpt 5224  ccom 5673  wf 6533  cfv 6537  (class class class)co 7405  Topctop 22750  TopOnctopon 22767   Cn ccn 23083  ko cxko 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-rest 17377  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cn 23086  df-cmp 23246  df-xko 23422
This theorem is referenced by:  cnmpt2k  23547
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