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Theorem dprdfcntz 16604
Description: A function on the elements of an internal direct product has pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdff.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
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 finSupp  .0.  }
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 2451 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 16601 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5657 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 16 . . 3  |-  ( ph  ->  F  Fn  I )
96ffvelrnda 5942 . . . . 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 5793 . . . . . . . . 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 5793 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =  z )  ->  ( F `  z )  =  ( F `  y ) )
1411, 13oveq12d 6208 . . . . . . . 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 dprdfcntz.z . . . . . . . . . . 11  |-  Z  =  (Cntz `  G )
2115, 16, 17, 18, 19, 20dprdcntz 16597 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( S `  y )  C_  ( Z `  ( S `  z )
) )
221, 2, 3, 4dprdfcl 16602 . . . . . . . . . . 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 3455 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  I )  /\  z  e.  I
)  /\  y  =/=  z )  ->  ( F `  y )  e.  ( Z `  ( S `  z )
) )
251, 2, 3, 4dprdfcl 16602 . . . . . . . . . . 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 15949 . . . . . . . . 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 6198 . . . . . . . . 9  |-  ( x  =  ( F `  z )  ->  (
( F `  y
) ( +g  `  G
) x )  =  ( ( F `  y ) ( +g  `  G ) ( F `
 z ) ) )
35 oveq1 6197 . . . . . . . . 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 5945 . . . . . . 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 5663 . . . . . . . 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 15942 . . . . . 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 5968 . . 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 5663 . 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    C_ wss 3426   class class class wbr 4390   dom cdm 4938   ran crn 4939    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   X_cixp 7363   finSupp cfsupp 7721   Basecbs 14276   +g cplusg 14340  Cntzccntz 15935   DProd cdprd 16580
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-ixp 7364  df-subg 15780  df-cntz 15937  df-dprd 16582
This theorem is referenced by:  dprdssv  16611  dprdfinv  16614  dprdfadd  16615  dprdfeq0  16617  dprdlub  16628  dmdprdsplitlem  16639  dpjidcl  16662
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