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Theorem cnmptk1p 23709
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 2735 . . . 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 23273 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐵):𝑋𝑌)
73, 4, 5, 6syl3anc 1370 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑌)
87fvmptelcdm 7133 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
92eleq1d 2824 . . . . 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 23497 . . . . . . . . . . 11 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
16 topontop 22935 . . . . . . . . . . 11 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
18 eqid 2735 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1918xkotopon 23624 . . . . . . . . . 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 23273 . . . . . . . . 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 1370 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2423fvmptelcdm 7133 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 23273 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
2610, 12, 24, 25syl3anc 1370 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
271fmpt 7130 . . . . . 6 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2826, 27sylibr 234 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
299, 28, 8rspcdva 3623 . . . 4 ((𝜑𝑥𝑋) → 𝐶𝑍)
301, 2, 8, 29fvmptd3 7039 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3130mpteq2dva 5248 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
32 eqid 2735 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33 toponuni 22936 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
344, 33syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
35 mpoeq12 7506 . . . . 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 2735 . . . . . 6 𝐾 = 𝐾
38 eqid 2735 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
3937, 38xkofvcn 23708 . . . . 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 2839 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
42 fveq1 6906 . . . 4 (𝑓 = (𝑦𝑌𝐴) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝑧))
43 fveq2 6907 . . . 4 (𝑧 = 𝐵 → ((𝑦𝑌𝐴)‘𝑧) = ((𝑦𝑌𝐴)‘𝐵))
4442, 43sylan9eq 2795 . . 3 ((𝑓 = (𝑦𝑌𝐴) ∧ 𝑧 = 𝐵) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝐵))
453, 21, 5, 20, 4, 41, 44cnmpt12 23691 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
4631, 45eqeltrrd 2840 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-nlly 23491  df-tx 23586  df-xko 23587
This theorem is referenced by:  xkohmeo  23839
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