MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt1k Structured version   Visualization version   Unicode version

Theorem cnmpt1k 20774
Description: The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmptk1.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
cnmptk1.l  |-  ( ph  ->  L  e.  (TopOn `  Z ) )
cnmpt1k.m  |-  ( ph  ->  M  e.  (TopOn `  W ) )
cnmpt1k.a  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )
cnmpt1k.b  |-  ( ph  ->  ( y  e.  Y  |->  ( z  e.  Z  |->  B ) )  e.  ( K  Cn  ( M  ^ko  L ) ) )
cnmpt1k.c  |-  ( z  =  A  ->  B  =  C )
Assertion
Ref Expression
cnmpt1k  |-  ( ph  ->  ( y  e.  Y  |->  ( x  e.  X  |->  C ) )  e.  ( K  Cn  ( M  ^ko  J ) ) )
Distinct variable groups:    x, y, J    x, K, y    x, L, y    x, M, y   
x, z, Z, y   
z, A    x, B    ph, x, y    x, X, y    x, Y, y   
z, C    y, A
Allowed substitution hints:    ph( z)    A( x)    B( y, z)    C( x, y)    J( z)    K( z)    L( z)    M( z)    W( x, y, z)    X( z)    Y( z)

Proof of Theorem cnmpt1k
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 cnmptk1.j . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 cnmptk1.l . . . . . . 7  |-  ( ph  ->  L  e.  (TopOn `  Z ) )
3 cnmpt1k.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )
4 cnf2 20342 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  (TopOn `  Z )  /\  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )  ->  ( x  e.  X  |->  A ) : X --> Z )
51, 2, 3, 4syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> Z )
6 eqid 2471 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
76fmpt 6058 . . . . . 6  |-  ( A. x  e.  X  A  e.  Z  <->  ( x  e.  X  |->  A ) : X --> Z )
85, 7sylibr 217 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  Z )
98adantr 472 . . . 4  |-  ( (
ph  /\  y  e.  Y )  ->  A. x  e.  X  A  e.  Z )
10 eqidd 2472 . . . 4  |-  ( (
ph  /\  y  e.  Y )  ->  (
x  e.  X  |->  A )  =  ( x  e.  X  |->  A ) )
11 eqidd 2472 . . . 4  |-  ( (
ph  /\  y  e.  Y )  ->  (
z  e.  Z  |->  B )  =  ( z  e.  Z  |->  B ) )
12 cnmpt1k.c . . . 4  |-  ( z  =  A  ->  B  =  C )
139, 10, 11, 12fmptcof 6073 . . 3  |-  ( (
ph  /\  y  e.  Y )  ->  (
( z  e.  Z  |->  B )  o.  (
x  e.  X  |->  A ) )  =  ( x  e.  X  |->  C ) )
1413mpteq2dva 4482 . 2  |-  ( ph  ->  ( y  e.  Y  |->  ( ( z  e.  Z  |->  B )  o.  ( x  e.  X  |->  A ) ) )  =  ( y  e.  Y  |->  ( x  e.  X  |->  C ) ) )
15 cnmptk1.k . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
16 cnmpt1k.b . . 3  |-  ( ph  ->  ( y  e.  Y  |->  ( z  e.  Z  |->  B ) )  e.  ( K  Cn  ( M  ^ko  L ) ) )
17 topontop 20018 . . . . 5  |-  ( L  e.  (TopOn `  Z
)  ->  L  e.  Top )
182, 17syl 17 . . . 4  |-  ( ph  ->  L  e.  Top )
19 cnmpt1k.m . . . . 5  |-  ( ph  ->  M  e.  (TopOn `  W ) )
20 topontop 20018 . . . . 5  |-  ( M  e.  (TopOn `  W
)  ->  M  e.  Top )
2119, 20syl 17 . . . 4  |-  ( ph  ->  M  e.  Top )
22 eqid 2471 . . . . 5  |-  ( M  ^ko  L )  =  ( M  ^ko  L )
2322xkotopon 20692 . . . 4  |-  ( ( L  e.  Top  /\  M  e.  Top )  ->  ( M  ^ko  L )  e.  (TopOn `  ( L  Cn  M
) ) )
2418, 21, 23syl2anc 673 . . 3  |-  ( ph  ->  ( M  ^ko  L )  e.  (TopOn `  ( L  Cn  M
) ) )
2521, 3xkoco1cn 20749 . . 3  |-  ( ph  ->  ( w  e.  ( L  Cn  M ) 
|->  ( w  o.  (
x  e.  X  |->  A ) ) )  e.  ( ( M  ^ko  L )  Cn  ( M  ^ko  J ) ) )
26 coeq1 4997 . . 3  |-  ( w  =  ( z  e.  Z  |->  B )  -> 
( w  o.  (
x  e.  X  |->  A ) )  =  ( ( z  e.  Z  |->  B )  o.  (
x  e.  X  |->  A ) ) )
2715, 16, 24, 25, 26cnmpt11 20755 . 2  |-  ( ph  ->  ( y  e.  Y  |->  ( ( z  e.  Z  |->  B )  o.  ( x  e.  X  |->  A ) ) )  e.  ( K  Cn  ( M  ^ko  J ) ) )
2814, 27eqeltrrd 2550 1  |-  ( ph  ->  ( y  e.  Y  |->  ( x  e.  X  |->  C ) )  e.  ( K  Cn  ( M  ^ko  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    |-> cmpt 4454    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995    Cn ccn 20317    ^ko cxko 20653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-cmp 20479  df-xko 20655
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator