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Theorem cnmptk1p 21298
 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 (𝐿 ^ko 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 cnmptk1p.j . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmptk1p.k . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 cnmptk1p.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))
4 cnf2 20863 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐵):𝑋𝑌)
51, 2, 3, 4syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋𝑌)
6 eqid 2610 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
76fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐵𝑌 ↔ (𝑥𝑋𝐵):𝑋𝑌)
85, 7sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵𝑌)
98r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
102adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
11 cnmptk1p.l . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘𝑍))
1211adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
13 cnmptk1p.n . . . . . . . . . . . 12 (𝜑𝐾 ∈ 𝑛-Locally Comp)
14 nllytop 21086 . . . . . . . . . . . 12 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . . 11 (𝜑𝐾 ∈ Top)
16 topontop 20541 . . . . . . . . . . . 12 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . . 11 (𝜑𝐿 ∈ Top)
18 eqid 2610 . . . . . . . . . . . 12 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
1918xkotopon 21213 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2015, 17, 19syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
21 cnmptk1p.a . . . . . . . . . 10 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
22 cnf2 20863 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
231, 20, 21, 22syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
24 eqid 2610 . . . . . . . . . 10 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
2524fmpt 6289 . . . . . . . . 9 (∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2623, 25sylibr 223 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
2726r19.21bi 2916 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
28 cnf2 20863 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
2910, 12, 27, 28syl3anc 1318 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
30 eqid 2610 . . . . . . 7 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
3130fmpt 6289 . . . . . 6 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
3229, 31sylibr 223 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
33 cnmptk1p.c . . . . . . 7 (𝑦 = 𝐵𝐴 = 𝐶)
3433eleq1d 2672 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑍𝐶𝑍))
3534rspcv 3278 . . . . 5 (𝐵𝑌 → (∀𝑦𝑌 𝐴𝑍𝐶𝑍))
369, 32, 35sylc 63 . . . 4 ((𝜑𝑥𝑋) → 𝐶𝑍)
3733, 30fvmptg 6189 . . . 4 ((𝐵𝑌𝐶𝑍) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
389, 36, 37syl2anc 691 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3938mpteq2dva 4672 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
40 eqid 2610 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
41 toponuni 20542 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
422, 41syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
43 mpt2eq12 6613 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
4440, 42, 43sylancr 694 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
45 eqid 2610 . . . . . 6 𝐾 = 𝐾
46 eqid 2610 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
4745, 46xkofvcn 21297 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
4813, 17, 47syl2anc 691 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
4944, 48eqeltrd 2688 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
50 fveq1 6102 . . . 4 (𝑓 = (𝑦𝑌𝐴) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝑧))
51 fveq2 6103 . . . 4 (𝑧 = 𝐵 → ((𝑦𝑌𝐴)‘𝑧) = ((𝑦𝑌𝐴)‘𝐵))
5250, 51sylan9eq 2664 . . 3 ((𝑓 = (𝑦𝑌𝐴) ∧ 𝑧 = 𝐵) → (𝑓𝑧) = ((𝑦𝑌𝐴)‘𝐵))
531, 21, 3, 20, 2, 49, 52cnmpt12 21280 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
5439, 53eqeltrrd 2689 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999  𝑛-Locally cnlly 21078   ×t ctx 21173   ^ko cxko 21174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-ntr 20634  df-nei 20712  df-cn 20841  df-cnp 20842  df-cmp 21000  df-nlly 21080  df-tx 21175  df-xko 21176 This theorem is referenced by:  xkohmeo  21428
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