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Theorem cnmptk1p 22385
 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 2758 . . . 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 21949 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐵):𝑋𝑌)
73, 4, 5, 6syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑌)
87fvmptelrn 6868 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
92eleq1d 2836 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑍𝐶𝑍))
104adantr 484 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
11 cnmptk1p.l . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘𝑍))
1211adantr 484 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
13 cnmptk1p.n . . . . . . . . . . 11 (𝜑𝐾 ∈ 𝑛-Locally Comp)
14 nllytop 22173 . . . . . . . . . . 11 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
16 topontop 21613 . . . . . . . . . . 11 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
18 eqid 2758 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1918xkotopon 22300 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2015, 17, 19syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
21 cnmptk1p.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
22 cnf2 21949 . . . . . . . . 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 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2423fvmptelrn 6868 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 21949 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
2610, 12, 24, 25syl3anc 1368 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
271fmpt 6865 . . . . . 6 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2826, 27sylibr 237 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
299, 28, 8rspcdva 3543 . . . 4 ((𝜑𝑥𝑋) → 𝐶𝑍)
301, 2, 8, 29fvmptd3 6782 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3130mpteq2dva 5127 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
32 eqid 2758 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33 toponuni 21614 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
344, 33syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
35 mpoeq12 7221 . . . . 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 590 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
37 eqid 2758 . . . . . 6 𝐾 = 𝐾
38 eqid 2758 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
3937, 38xkofvcn 22384 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
4013, 17, 39syl2anc 587 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
4136, 40eqeltrd 2852 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
42 fveq1 6657 . . . 4 (𝑓 = (𝑦𝑌𝐴) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝑧))
43 fveq2 6658 . . . 4 (𝑧 = 𝐵 → ((𝑦𝑌𝐴)‘𝑧) = ((𝑦𝑌𝐴)‘𝐵))
4442, 43sylan9eq 2813 . . 3 ((𝑓 = (𝑦𝑌𝐴) ∧ 𝑧 = 𝐵) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝐵))
453, 21, 5, 20, 4, 41, 44cnmpt12 22367 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
4631, 45eqeltrrd 2853 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∪ cuni 4798   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  Topctop 21593  TopOnctopon 21610   Cn ccn 21924  Compccmp 22086  𝑛-Locally cnlly 22165   ×t ctx 22260   ↑ko cxko 22261 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-fin 8531  df-fi 8908  df-rest 16754  df-topgen 16775  df-pt 16776  df-top 21594  df-topon 21611  df-bases 21646  df-ntr 21720  df-nei 21798  df-cn 21927  df-cnp 21928  df-cmp 22087  df-nlly 22167  df-tx 22262  df-xko 22263 This theorem is referenced by:  xkohmeo  22515
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