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Theorem cnmptk1p 23694
Description: The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1p.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1p.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1p.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptk1p.n (𝜑𝐾 ∈ 𝑛-Locally Comp)
cnmptk1p.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
cnmptk1p.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))
cnmptk1p.c (𝑦 = 𝐵𝐴 = 𝐶)
Assertion
Ref Expression
cnmptk1p (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝑦,𝐵   𝑦,𝐶   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑦,𝑍
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(𝑥)

Proof of Theorem cnmptk1p
Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2 cnmptk1p.c . . . 4 (𝑦 = 𝐵𝐴 = 𝐶)
3 cnmptk1p.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmptk1p.k . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 cnmptk1p.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))
6 cnf2 23258 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐵):𝑋𝑌)
73, 4, 5, 6syl3anc 1372 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑌)
87fvmptelcdm 7132 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
92eleq1d 2825 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑍𝐶𝑍))
104adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
11 cnmptk1p.l . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘𝑍))
1211adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
13 cnmptk1p.n . . . . . . . . . . 11 (𝜑𝐾 ∈ 𝑛-Locally Comp)
14 nllytop 23482 . . . . . . . . . . 11 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
16 topontop 22920 . . . . . . . . . . 11 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
18 eqid 2736 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1918xkotopon 23609 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2015, 17, 19syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
21 cnmptk1p.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
22 cnf2 23258 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
233, 20, 21, 22syl3anc 1372 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2423fvmptelcdm 7132 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 23258 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
2610, 12, 24, 25syl3anc 1372 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
271fmpt 7129 . . . . . 6 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2826, 27sylibr 234 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
299, 28, 8rspcdva 3622 . . . 4 ((𝜑𝑥𝑋) → 𝐶𝑍)
301, 2, 8, 29fvmptd3 7038 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3130mpteq2dva 5241 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
32 eqid 2736 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33 toponuni 22921 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
344, 33syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
35 mpoeq12 7507 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
3632, 34, 35sylancr 587 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
37 eqid 2736 . . . . . 6 𝐾 = 𝐾
38 eqid 2736 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
3937, 38xkofvcn 23693 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
4013, 17, 39syl2anc 584 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
4136, 40eqeltrd 2840 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
42 fveq1 6904 . . . 4 (𝑓 = (𝑦𝑌𝐴) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝑧))
43 fveq2 6905 . . . 4 (𝑧 = 𝐵 → ((𝑦𝑌𝐴)‘𝑧) = ((𝑦𝑌𝐴)‘𝐵))
4442, 43sylan9eq 2796 . . 3 ((𝑓 = (𝑦𝑌𝐴) ∧ 𝑧 = 𝐵) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝐵))
453, 21, 5, 20, 4, 41, 44cnmpt12 23676 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
4631, 45eqeltrrd 2841 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060   cuni 4906  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  Topctop 22900  TopOnctopon 22917   Cn ccn 23233  Compccmp 23395  𝑛-Locally cnlly 23474   ×t ctx 23569  ko cxko 23570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-pt 17490  df-top 22901  df-topon 22918  df-bases 22954  df-ntr 23029  df-nei 23107  df-cn 23236  df-cnp 23237  df-cmp 23396  df-nlly 23476  df-tx 23571  df-xko 23572
This theorem is referenced by:  xkohmeo  23824
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