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Theorem cnmpt1k 19373
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 18971 . . . . . . 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 1219 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> Z )
6 eqid 2451 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
76fmpt 5965 . . . . . 6  |-  ( A. x  e.  X  A  e.  Z  <->  ( x  e.  X  |->  A ) : X --> Z )
85, 7sylibr 212 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  Z )
98adantr 465 . . . 4  |-  ( (
ph  /\  y  e.  Y )  ->  A. x  e.  X  A  e.  Z )
10 eqidd 2452 . . . 4  |-  ( (
ph  /\  y  e.  Y )  ->  (
x  e.  X  |->  A )  =  ( x  e.  X  |->  A ) )
11 eqidd 2452 . . . 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 5978 . . 3  |-  ( (
ph  /\  y  e.  Y )  ->  (
( z  e.  Z  |->  B )  o.  (
x  e.  X  |->  A ) )  =  ( x  e.  X  |->  C ) )
1413mpteq2dva 4478 . 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 18649 . . . . 5  |-  ( L  e.  (TopOn `  Z
)  ->  L  e.  Top )
182, 17syl 16 . . . 4  |-  ( ph  ->  L  e.  Top )
19 cnmpt1k.m . . . . 5  |-  ( ph  ->  M  e.  (TopOn `  W ) )
20 topontop 18649 . . . . 5  |-  ( M  e.  (TopOn `  W
)  ->  M  e.  Top )
2119, 20syl 16 . . . 4  |-  ( ph  ->  M  e.  Top )
22 eqid 2451 . . . . 5  |-  ( M  ^ko  L )  =  ( M  ^ko  L )
2322xkotopon 19291 . . . 4  |-  ( ( L  e.  Top  /\  M  e.  Top )  ->  ( M  ^ko  L )  e.  (TopOn `  ( L  Cn  M
) ) )
2418, 21, 23syl2anc 661 . . 3  |-  ( ph  ->  ( M  ^ko  L )  e.  (TopOn `  ( L  Cn  M
) ) )
2521, 3xkoco1cn 19348 . . 3  |-  ( ph  ->  ( w  e.  ( L  Cn  M ) 
|->  ( w  o.  (
x  e.  X  |->  A ) ) )  e.  ( ( M  ^ko  L )  Cn  ( M  ^ko  J ) ) )
26 coeq1 5097 . . 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 19354 . 2  |-  ( ph  ->  ( y  e.  Y  |->  ( ( z  e.  Z  |->  B )  o.  ( x  e.  X  |->  A ) ) )  e.  ( K  Cn  ( M  ^ko  J ) ) )
2814, 27eqeltrrd 2540 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 369    = wceq 1370    e. wcel 1758   A.wral 2795    |-> cmpt 4450    o. ccom 4944   -->wf 5514   ` cfv 5518  (class class class)co 6192   Topctop 18616  TopOnctopon 18617    Cn ccn 18946    ^ko cxko 19252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-fin 7416  df-fi 7764  df-rest 14465  df-topgen 14486  df-top 18621  df-bases 18623  df-topon 18624  df-cn 18949  df-cmp 19108  df-xko 19254
This theorem is referenced by: (None)
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