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Theorem dprdfcntz 15250
Description: A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdff.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
dprdff.1  |-  ( ph  ->  G dom DProd  S )
dprdff.2  |-  ( ph  ->  dom  S  =  I )
dprdff.3  |-  ( ph  ->  F  e.  W )
dprdfcntz.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
dprdfcntz  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Distinct variable groups:    h, F    h, i, I    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    G( h, i)    W( h, i)    .0. ( i)    Z( h, i)

Proof of Theorem dprdfcntz
Dummy variables  y 
z  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdff.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2 dprdff.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
3 dprdff.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
4 dprdff.3 . . . . 5  |-  ( ph  ->  F  e.  W )
5 eqid 2283 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15247 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5389 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 15 . . 3  |-  ( ph  ->  F  Fn  I )
9 ffvelrn 5663 . . . . . 6  |-  ( ( F : I --> ( Base `  G )  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  G
) )
106, 9sylan 457 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  G
) )
11 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  y  =  z )
1211fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  y )  =  ( F `  z ) )
1311eqcomd 2288 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  z  =  y )
1413fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  z )  =  ( F `  y ) )
1512, 14oveq12d 5876 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
162ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  G dom DProd  S )
173ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  dom  S  =  I )
18 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  y  e.  I )
19 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  z  e.  I )
20 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  y  =/=  z )
21 dprdfcntz.z . . . . . . . . . . 11  |-  Z  =  (Cntz `  G )
2216, 17, 18, 19, 20, 21dprdcntz 15243 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( S `  y )  C_  ( Z `  ( S `  z )
) )
231, 2, 3, 4dprdfcl 15248 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
2423ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( S `  y
) )
2522, 24sseldd 3181 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( Z `  ( S `  z )
) )
261, 2, 3, 4dprdfcl 15248 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2726adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2827adantr 451 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  z )  e.  ( S `  z
) )
29 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3029, 21cntzi 14805 . . . . . . . . 9  |-  ( ( ( F `  y
)  e.  ( Z `
 ( S `  z ) )  /\  ( F `  z )  e.  ( S `  z ) )  -> 
( ( F `  y ) ( +g  `  G ) ( F `
 z ) )  =  ( ( F `
 z ) ( +g  `  G ) ( F `  y
) ) )
3125, 28, 30syl2anc 642 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3215, 31pm2.61dane 2524 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3332ralrimiva 2626 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  A. z  e.  I  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) )
348adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  Fn  I )
35 oveq2 5866 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
( F `  y
) ( +g  `  G
) x )  =  ( ( F `  y ) ( +g  `  G ) ( F `
 z ) ) )
36 oveq1 5865 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  G
) ( F `  y ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3735, 36eqeq12d 2297 . . . . . . . 8  |-  ( x  =  ( F `  z )  ->  (
( ( F `  y ) ( +g  `  G ) x )  =  ( x ( +g  `  G ) ( F `  y
) )  <->  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) ) )
3837ralrn 5668 . . . . . . 7  |-  ( F  Fn  I  ->  ( A. x  e.  ran  F ( ( F `  y ) ( +g  `  G ) x )  =  ( x ( +g  `  G ) ( F `  y
) )  <->  A. z  e.  I  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) ) )
3934, 38syl 15 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( A. x  e.  ran  F ( ( F `  y ) ( +g  `  G ) x )  =  ( x ( +g  `  G ) ( F `  y
) )  <->  A. z  e.  I  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) ) )
4033, 39mpbird 223 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  A. x  e.  ran  F ( ( F `  y ) ( +g  `  G
) x )  =  ( x ( +g  `  G ) ( F `
 y ) ) )
41 frn 5395 . . . . . . . 8  |-  ( F : I --> ( Base `  G )  ->  ran  F 
C_  ( Base `  G
) )
426, 41syl 15 . . . . . . 7  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
4342adantr 451 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ran  F 
C_  ( Base `  G
) )
445, 29, 21elcntz 14798 . . . . . 6  |-  ( ran 
F  C_  ( Base `  G )  ->  (
( F `  y
)  e.  ( Z `
 ran  F )  <->  ( ( F `  y
)  e.  ( Base `  G )  /\  A. x  e.  ran  F ( ( F `  y
) ( +g  `  G
) x )  =  ( x ( +g  `  G ) ( F `
 y ) ) ) ) )
4543, 44syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( F `  y
)  e.  ( Z `
 ran  F )  <->  ( ( F `  y
)  e.  ( Base `  G )  /\  A. x  e.  ran  F ( ( F `  y
) ( +g  `  G
) x )  =  ( x ( +g  `  G ) ( F `
 y ) ) ) ) )
4610, 40, 45mpbir2and 888 . . . 4  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Z `  ran  F ) )
4746ralrimiva 2626 . . 3  |-  ( ph  ->  A. y  e.  I 
( F `  y
)  e.  ( Z `
 ran  F )
)
48 ffnfv 5685 . . 3  |-  ( F : I --> ( Z `
 ran  F )  <->  ( F  Fn  I  /\  A. y  e.  I  ( F `  y )  e.  ( Z `  ran  F ) ) )
498, 47, 48sylanbrc 645 . 2  |-  ( ph  ->  F : I --> ( Z `
 ran  F )
)
50 frn 5395 . 2  |-  ( F : I --> ( Z `
 ran  F )  ->  ran  F  C_  ( Z `  ran  F ) )
5149, 50syl 15 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdssv  15251  dprdfinv  15254  dprdfadd  15255  dprdfeq0  15257  dprdlub  15261  dmdprdsplitlem  15272  dpjidcl  15293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-subg 14618  df-cntz 14793  df-dprd 15233
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