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Theorem dprdfcntz 15266
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 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15263 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5405 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 15 . . 3  |-  ( ph  ->  F  Fn  I )
9 ffvelrn 5679 . . . . . 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 5545 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  y )  =  ( F `  z ) )
1311eqcomd 2301 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  z  =  y )
1413fveq2d 5545 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  z )  =  ( F `  y ) )
1512, 14oveq12d 5892 . . . . . . . 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 15259 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( S `  y )  C_  ( Z `  ( S `  z )
) )
231, 2, 3, 4dprdfcl 15264 . . . . . . . . . . 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 3194 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( Z `  ( S `  z )
) )
261, 2, 3, 4dprdfcl 15264 . . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
3029, 21cntzi 14821 . . . . . . . . 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 2537 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  I )  /\  z  e.  I )  ->  (
( F `  y
) ( +g  `  G
) ( F `  z ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3332ralrimiva 2639 . . . . . 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 5882 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
( F `  y
) ( +g  `  G
) x )  =  ( ( F `  y ) ( +g  `  G ) ( F `
 z ) ) )
36 oveq1 5881 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  G
) ( F `  y ) )  =  ( ( F `  z ) ( +g  `  G ) ( F `
 y ) ) )
3735, 36eqeq12d 2310 . . . . . . . 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 5684 . . . . . . 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 5411 . . . . . . . 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 14814 . . . . . 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 2639 . . 3  |-  ( ph  ->  A. y  e.  I 
( F `  y
)  e.  ( Z `
 ran  F )
)
48 ffnfv 5701 . . 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 5411 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Fincfn 6879   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdssv  15267  dprdfinv  15270  dprdfadd  15271  dprdfeq0  15273  dprdlub  15277  dmdprdsplitlem  15288  dpjidcl  15309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-subg 14634  df-cntz 14809  df-dprd 15249
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