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Theorem cnmpt1k 21466
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 21034 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐴):𝑋𝑍)
51, 2, 3, 4syl3anc 1324 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑍)
6 eqid 2620 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6367 . . . . . 6 (∀𝑥𝑋 𝐴𝑍 ↔ (𝑥𝑋𝐴):𝑋𝑍)
85, 7sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑍)
98adantr 481 . . . 4 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴𝑍)
10 eqidd 2621 . . . 4 ((𝜑𝑦𝑌) → (𝑥𝑋𝐴) = (𝑥𝑋𝐴))
11 eqidd 2621 . . . 4 ((𝜑𝑦𝑌) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
12 cnmpt1k.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
139, 10, 11, 12fmptcof 6383 . . 3 ((𝜑𝑦𝑌) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
1413mpteq2dva 4735 . 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 20699 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
182, 17syl 17 . . . 4 (𝜑𝐿 ∈ Top)
19 cnmpt1k.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
20 topontop 20699 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
2119, 20syl 17 . . . 4 (𝜑𝑀 ∈ Top)
22 eqid 2620 . . . . 5 (𝑀 ^ko 𝐿) = (𝑀 ^ko 𝐿)
2322xkotopon 21384 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2418, 21, 23syl2anc 692 . . 3 (𝜑 → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
2521, 3xkoco1cn 21441 . . 3 (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥𝑋𝐴))) ∈ ((𝑀 ^ko 𝐿) Cn (𝑀 ^ko 𝐽)))
26 coeq1 5268 . . 3 (𝑤 = (𝑧𝑍𝐵) → (𝑤 ∘ (𝑥𝑋𝐴)) = ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴)))
2715, 16, 24, 25, 26cnmpt11 21447 . 2 (𝜑 → (𝑦𝑌 ↦ ((𝑧𝑍𝐵) ∘ (𝑥𝑋𝐴))) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
2814, 27eqeltrrd 2700 1 (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  cmpt 4720  ccom 5108  wf 5872  cfv 5876  (class class class)co 6635  Topctop 20679  TopOnctopon 20696   Cn ccn 21009   ^ko cxko 21345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-fin 7944  df-fi 8302  df-rest 16064  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cn 21012  df-cmp 21171  df-xko 21347
This theorem is referenced by: (None)
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