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Theorem dprdfcntzOLD 16612
Description: A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdfcntz 16606 as of 11-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dprdffOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
dprdffOLD.1  |-  ( ph  ->  G dom DProd  S )
dprdffOLD.2  |-  ( ph  ->  dom  S  =  I )
dprdffOLD.3  |-  ( ph  ->  F  e.  W )
dprdfcntzOLD.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
dprdfcntzOLD  |-  ( 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 dprdfcntzOLD
Dummy variables  y 
z  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdffOLD.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2 dprdffOLD.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
3 dprdffOLD.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
4 dprdffOLD.3 . . . . 5  |-  ( ph  ->  F  e.  W )
5 eqid 2451 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdffOLD 16609 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5659 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 16 . . 3  |-  ( ph  ->  F  Fn  I )
96ffvelrnda 5944 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  G
) )
10 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  y  =  z )
1110fveq2d 5795 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  y )  =  ( F `  z ) )
1210eqcomd 2459 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  z  =  y )
1312fveq2d 5795 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  z )  =  ( F `  y ) )
1411, 13oveq12d 6210 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
152ad3antrrr 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  G dom DProd  S )
163ad3antrrr 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  dom  S  =  I )
17 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  y  e.  I )
18 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  z  e.  I )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  y  =/=  z )
20 dprdfcntzOLD.z . . . . . . . . . . 11  |-  Z  =  (Cntz `  G )
2115, 16, 17, 18, 19, 20dprdcntz 16599 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( S `  y )  C_  ( Z `  ( S `  z )
) )
221, 2, 3, 4dprdfclOLD 16610 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
2322ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( S `  y
) )
2421, 23sseldd 3457 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( Z `  ( S `  z )
) )
251, 2, 3, 4dprdfclOLD 16610 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2625adantlr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2726adantr 465 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  z )  e.  ( S `  z
) )
28 eqid 2451 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
2928, 20cntzi 15951 . . . . . . . . 9  |-  ( ( ( F `  y
)  e.  ( Z `
 ( S `  z ) )  /\  ( F `  z )  e.  ( S `  z ) )  -> 
( ( F `  y ) ( +g  `  G ) ( F `
 z ) )  =  ( ( F `
 z ) ( +g  `  G ) ( F `  y
) ) )
3024, 27, 29syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3114, 30pm2.61dane 2766 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3231ralrimiva 2822 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  A. z  e.  I  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) )
338adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  Fn  I )
34 oveq2 6200 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
( F `  y
) ( +g  `  G
) x )  =  ( ( F `  y ) ( +g  `  G ) ( F `
 z ) ) )
35 oveq1 6199 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  G
) ( F `  y ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3634, 35eqeq12d 2473 . . . . . . . 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 ) ) ) )
3736ralrn 5947 . . . . . . 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 ) ) ) )
3833, 37syl 16 . . . . . 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 ) ) ) )
3932, 38mpbird 232 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  A. x  e.  ran  F ( ( F `  y ) ( +g  `  G
) x )  =  ( x ( +g  `  G ) ( F `
 y ) ) )
40 frn 5665 . . . . . . . 8  |-  ( F : I --> ( Base `  G )  ->  ran  F 
C_  ( Base `  G
) )
416, 40syl 16 . . . . . . 7  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
4241adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ran  F 
C_  ( Base `  G
) )
435, 28, 20elcntz 15944 . . . . . 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 ) ) ) ) )
4442, 43syl 16 . . . . 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 ) ) ) ) )
459, 39, 44mpbir2and 913 . . . 4  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Z `  ran  F ) )
4645ralrimiva 2822 . . 3  |-  ( ph  ->  A. y  e.  I 
( F `  y
)  e.  ( Z `
 ran  F )
)
47 ffnfv 5970 . . 3  |-  ( F : I --> ( Z `
 ran  F )  <->  ( F  Fn  I  /\  A. y  e.  I  ( F `  y )  e.  ( Z `  ran  F ) ) )
488, 46, 47sylanbrc 664 . 2  |-  ( ph  ->  F : I --> ( Z `
 ran  F )
)
49 frn 5665 . 2  |-  ( F : I --> ( Z `
 ran  F )  ->  ran  F  C_  ( Z `  ran  F ) )
5048, 49syl 16 1  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   {crab 2799   _Vcvv 3070    \ cdif 3425    C_ wss 3428   {csn 3977   class class class wbr 4392   `'ccnv 4939   dom cdm 4940   ran crn 4941   "cima 4943    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192   X_cixp 7365   Fincfn 7412   Basecbs 14278   +g cplusg 14342  Cntzccntz 15937   DProd cdprd 16582
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-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-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  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-1st 6679  df-2nd 6680  df-ixp 7366  df-subg 15782  df-cntz 15939  df-dprd 16584
This theorem is referenced by:  dprdfinvOLD  16623  dprdfaddOLD  16624  dprdfeq0OLD  16626  dmdprdsplitlemOLD  16642  dpjidclOLD  16671
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