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Theorem dprdfcntzOLD 17253
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 17247 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 2454 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdffOLD 17250 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5713 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 16 . . 3  |-  ( ph  ->  F  Fn  I )
96ffvelrnda 6007 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  G
) )
10 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  y  =  z )
1110fveq2d 5852 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  y )  =  ( F `  z ) )
1210eqcomd 2462 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  z  =  y )
1312fveq2d 5852 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  z )  =  ( F `  y ) )
1411, 13oveq12d 6288 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
152ad3antrrr 727 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  G dom DProd  S )
163ad3antrrr 727 . . . . . . . . . . 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 753 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  z  e.  I )
19 simpr 459 . . . . . . . . . . 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 17239 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( S `  y )  C_  ( Z `  ( S `  z )
) )
221, 2, 3, 4dprdfclOLD 17251 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( S `  y
) )
2322ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( S `  y
) )
2421, 23sseldd 3490 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( Z `  ( S `  z )
) )
251, 2, 3, 4dprdfclOLD 17251 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2625adantlr 712 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  ( F `  z )  e.  ( S `  z
) )
2726adantr 463 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  z )  e.  ( S `  z
) )
28 eqid 2454 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
2928, 20cntzi 16569 . . . . . . . . 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 659 . . . . . . . 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 2772 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3231ralrimiva 2868 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  A. z  e.  I  ( ( F `  y )
( +g  `  G ) ( F `  z
) )  =  ( ( F `  z
) ( +g  `  G
) ( F `  y ) ) )
338adantr 463 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  Fn  I )
34 oveq2 6278 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
( F `  y
) ( +g  `  G
) x )  =  ( ( F `  y ) ( +g  `  G ) ( F `
 z ) ) )
35 oveq1 6277 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  G
) ( F `  y ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3634, 35eqeq12d 2476 . . . . . . . 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 6010 . . . . . . 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 5719 . . . . . . . 8  |-  ( F : I --> ( Base `  G )  ->  ran  F 
C_  ( Base `  G
) )
416, 40syl 16 . . . . . . 7  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
4241adantr 463 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ran  F 
C_  ( Base `  G
) )
435, 28, 20elcntz 16562 . . . . . 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 920 . . . 4  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Z `  ran  F ) )
4645ralrimiva 2868 . . 3  |-  ( ph  ->  A. y  e.  I 
( F `  y
)  e.  ( Z `
 ran  F )
)
47 ffnfv 6033 . . 3  |-  ( F : I --> ( Z `
 ran  F )  <->  ( F  Fn  I  /\  A. y  e.  I  ( F `  y )  e.  ( Z `  ran  F ) ) )
488, 46, 47sylanbrc 662 . 2  |-  ( ph  ->  F : I --> ( Z `
 ran  F )
)
49 frn 5719 . 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   class class class wbr 4439   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   X_cixp 7462   Fincfn 7509   Basecbs 14719   +g cplusg 14787  Cntzccntz 16555   DProd cdprd 17222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-ixp 7463  df-subg 16400  df-cntz 16557  df-dprd 17224
This theorem is referenced by:  dprdfinvOLD  17264  dprdfaddOLD  17265  dprdfeq0OLD  17267  dmdprdsplitlemOLD  17283  dpjidclOLD  17312
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