MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmptk1p Structured version   Visualization version   GIF version

Theorem cnmptk1p 23059
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmptk1p.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmptk1p.l (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
cnmptk1p.n (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)
cnmptk1p.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptk1p.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))
cnmptk1p.c (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)
Assertion
Ref Expression
cnmptk1p (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝐿   𝑦,𝐡   𝑦,𝐢   π‘₯,𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,π‘₯,𝑦   𝑦,𝑍
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯)   𝐢(π‘₯)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(π‘₯)

Proof of Theorem cnmptk1p
Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
2 cnmptk1p.c . . . 4 (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)
3 cnmptk1p.j . . . . . 6 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 cnmptk1p.k . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
5 cnmptk1p.b . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))
6 cnf2 22623 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆπ‘Œ)
73, 4, 5, 6syl3anc 1372 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆπ‘Œ)
87fvmptelcdm 7065 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ π‘Œ)
92eleq1d 2819 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴 ∈ 𝑍 ↔ 𝐢 ∈ 𝑍))
104adantr 482 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
11 cnmptk1p.l . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
1211adantr 482 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
13 cnmptk1p.n . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)
14 nllytop 22847 . . . . . . . . . . 11 (𝐾 ∈ 𝑛-Locally Comp β†’ 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ Top)
16 topontop 22285 . . . . . . . . . . 11 (𝐿 ∈ (TopOnβ€˜π‘) β†’ 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ Top)
18 eqid 2733 . . . . . . . . . . 11 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
1918xkotopon 22974 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
2015, 17, 19syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
21 cnmptk1p.a . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
22 cnf2 22623 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)) ∧ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
233, 20, 21, 22syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
2423fvmptelcdm 7065 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 22623 . . . . . . 7 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜π‘) ∧ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
2610, 12, 24, 25syl3anc 1372 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
271fmpt 7062 . . . . . 6 (βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
2826, 27sylibr 233 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍)
299, 28, 8rspcdva 3584 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ 𝑍)
301, 2, 8, 29fvmptd3 6975 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅) = 𝐢)
3130mpteq2dva 5209 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))
32 eqid 2733 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33 toponuni 22286 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
344, 33syl 17 . . . . 5 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
35 mpoeq12 7434 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ π‘Œ = βˆͺ 𝐾) β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)))
3632, 34, 35sylancr 588 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)))
37 eqid 2733 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
38 eqid 2733 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§))
3937, 38xkofvcn 23058 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
4013, 17, 39syl2anc 585 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
4136, 40eqeltrd 2834 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
42 fveq1 6845 . . . 4 (𝑓 = (𝑦 ∈ π‘Œ ↦ 𝐴) β†’ (π‘“β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§))
43 fveq2 6846 . . . 4 (𝑧 = 𝐡 β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅))
4442, 43sylan9eq 2793 . . 3 ((𝑓 = (𝑦 ∈ π‘Œ ↦ 𝐴) ∧ 𝑧 = 𝐡) β†’ (π‘“β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅))
453, 21, 5, 20, 4, 41, 44cnmpt12 23041 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅)) ∈ (𝐽 Cn 𝐿))
4631, 45eqeltrrd 2835 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆͺ cuni 4869   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Topctop 22265  TopOnctopon 22282   Cn ccn 22598  Compccmp 22760  π‘›-Locally cnlly 22839   Γ—t ctx 22934   ↑ko cxko 22935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-pt 17334  df-top 22266  df-topon 22283  df-bases 22319  df-ntr 22394  df-nei 22472  df-cn 22601  df-cnp 22602  df-cmp 22761  df-nlly 22841  df-tx 22936  df-xko 22937
This theorem is referenced by:  xkohmeo  23189
  Copyright terms: Public domain W3C validator