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Theorem cnmptk1 22277
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 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3 cnmptk1.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
43adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5 cnmptk1.j . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 21509 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
8 topontop 21509 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
93, 8syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
10 eqid 2824 . . . . . . . . . 10 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1110xkotopon 22196 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
127, 9, 11syl2anc 587 . . . . . . . 8 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
13 cnmptk1.a . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
14 cnf2 21845 . . . . . . . 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))
1615fvmptelrn 6860 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
17 cnf2 21845 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
182, 4, 16, 17syl3anc 1368 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
19 eqid 2824 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2019fmpt 6857 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2118, 20sylibr 237 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
22 eqidd 2825 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
23 eqidd 2825 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
24 cnmptk1.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2521, 22, 23, 24fmptcof 6875 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
2625mpteq2dva 5144 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
27 cnmptk1.b . . . 4 (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
287, 27xkoco2cn 22254 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ ((𝑧𝑍𝐵) ∘ 𝑤)) ∈ ((𝐿ko 𝐾) Cn (𝑀ko 𝐾)))
29 coeq2 5712 . . 3 (𝑤 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑤) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
305, 13, 12, 28, 29cnmpt11 22259 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀ko 𝐾)))
3126, 30eqeltrrd 2917 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3132  cmpt 5129  ccom 5542  wf 6334  cfv 6338  (class class class)co 7140  Topctop 21489  TopOnctopon 21506   Cn ccn 21820  ko cxko 22157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-iin 4905  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-fin 8498  df-fi 8861  df-rest 16687  df-topgen 16708  df-top 21490  df-topon 21507  df-bases 21542  df-cn 21823  df-cmp 21983  df-xko 22159
This theorem is referenced by:  cnmpt2k  22284
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