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Theorem dprdf1o 17665
Description: Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdf1o.1  |-  ( ph  ->  G dom DProd  S )
dprdf1o.2  |-  ( ph  ->  dom  S  =  I )
dprdf1o.3  |-  ( ph  ->  F : J -1-1-onto-> I )
Assertion
Ref Expression
dprdf1o  |-  ( ph  ->  ( G dom DProd  ( S  o.  F )  /\  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S ) ) )

Proof of Theorem dprdf1o
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2451 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2451 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdf1o.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
5 dprdgrp 17637 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
64, 5syl 17 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdf1o.3 . . . . 5  |-  ( ph  ->  F : J -1-1-onto-> I )
8 f1of1 5813 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J -1-1-> I )
97, 8syl 17 . . . 4  |-  ( ph  ->  F : J -1-1-> I
)
10 dprdf1o.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
114, 10dprddomcld 17633 . . . 4  |-  ( ph  ->  I  e.  _V )
12 f1dmex 6763 . . . 4  |-  ( ( F : J -1-1-> I  /\  I  e.  _V )  ->  J  e.  _V )
139, 11, 12syl2anc 667 . . 3  |-  ( ph  ->  J  e.  _V )
144, 10dprdf2 17639 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
15 f1of 5814 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J
--> I )
167, 15syl 17 . . . 4  |-  ( ph  ->  F : J --> I )
17 fco 5739 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  F : J --> I )  -> 
( S  o.  F
) : J --> (SubGrp `  G ) )
1814, 16, 17syl2anc 667 . . 3  |-  ( ph  ->  ( S  o.  F
) : J --> (SubGrp `  G ) )
194adantr 467 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  G dom DProd  S )
2010adantr 467 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  dom  S  =  I )
2116adantr 467 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J --> I )
22 simpr1 1014 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  e.  J )
2321, 22ffvelrnd 6023 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  e.  I )
24 simpr2 1015 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
y  e.  J )
2521, 24ffvelrnd 6023 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  y
)  e.  I )
26 simpr3 1016 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  =/=  y )
279adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J -1-1-> I )
28 f1fveq 6163 . . . . . . . 8  |-  ( ( F : J -1-1-> I  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
( F `  x
)  =  ( F `
 y )  <->  x  =  y ) )
2927, 22, 24, 28syl12anc 1266 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
3029necon3bid 2668 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =/=  ( F `  y )  <->  x  =/=  y ) )
3126, 30mpbird 236 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  =/=  ( F `
 y ) )
3219, 20, 23, 25, 31, 1dprdcntz 17640 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( S `  ( F `  x )
)  C_  ( (Cntz `  G ) `  ( S `  ( F `  y ) ) ) )
33 fvco3 5942 . . . . 5  |-  ( ( F : J --> I  /\  x  e.  J )  ->  ( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
3421, 22, 33syl2anc 667 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
35 fvco3 5942 . . . . . 6  |-  ( ( F : J --> I  /\  y  e.  J )  ->  ( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
3621, 24, 35syl2anc 667 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
3736fveq2d 5869 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( ( S  o.  F ) `  y ) )  =  ( (Cntz `  G
) `  ( S `  ( F `  y
) ) ) )
3832, 34, 373sstr4d 3475 . . 3  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  o.  F
) `  y )
) )
3916, 33sylan 474 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) `  x )  =  ( S `  ( F `  x ) ) )
40 imaco 5340 . . . . . . . . 9  |-  ( ( S  o.  F )
" ( J  \  { x } ) )  =  ( S
" ( F "
( J  \  {
x } ) ) )
417adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  F : J -1-1-onto-> I )
42 dff1o3 5820 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  <->  ( F : J -onto-> I  /\  Fun  `' F ) )
4342simprbi 466 . . . . . . . . . . . 12  |-  ( F : J -1-1-onto-> I  ->  Fun  `' F )
44 imadif 5658 . . . . . . . . . . . 12  |-  ( Fun  `' F  ->  ( F
" ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
4541, 43, 443syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
46 f1ofo 5821 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  ->  F : J -onto-> I )
47 foima 5798 . . . . . . . . . . . . 13  |-  ( F : J -onto-> I  -> 
( F " J
)  =  I )
4841, 46, 473syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " J )  =  I )
49 f1ofn 5815 . . . . . . . . . . . . . . 15  |-  ( F : J -1-1-onto-> I  ->  F  Fn  J )
507, 49syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  J )
51 fnsnfv 5925 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  J  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F " { x } ) )
5250, 51sylan 474 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F
" { x }
) )
5352eqcomd 2457 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " { x }
)  =  { ( F `  x ) } )
5448, 53difeq12d 3552 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " J
)  \  ( F " { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
5545, 54eqtrd 2485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
5655imaeq2d 5168 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( S " ( F "
( J  \  {
x } ) ) )  =  ( S
" ( I  \  { ( F `  x ) } ) ) )
5740, 56syl5eq 2497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) " ( J 
\  { x }
) )  =  ( S " ( I 
\  { ( F `
 x ) } ) ) )
5857unieqd 4208 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  U. (
( S  o.  F
) " ( J 
\  { x }
) )  =  U. ( S " ( I 
\  { ( F `
 x ) } ) ) )
5958fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  o.  F ) " ( J  \  { x }
) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { ( F `  x ) } ) ) ) )
6039, 59ineq12d 3635 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  ( ( S `  ( F `  x ) )  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) ) )
614adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  G dom DProd  S )
6210adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  dom  S  =  I )
6316ffvelrnda 6022 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F `  x )  e.  I )
6461, 62, 63, 2, 3dprddisj 17641 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( S `  ( F `  x )
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) )  =  { ( 0g `  G ) } )
6560, 64eqtrd 2485 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
66 eqimss 3484 . . . 4  |-  ( ( ( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) }  ->  ( ( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
6765, 66syl 17 . . 3  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
681, 2, 3, 6, 13, 18, 38, 67dmdprdd 17631 . 2  |-  ( ph  ->  G dom DProd  ( S  o.  F ) )
69 rnco2 5342 . . . . . 6  |-  ran  ( S  o.  F )  =  ( S " ran  F )
70 forn 5796 . . . . . . . . 9  |-  ( F : J -onto-> I  ->  ran  F  =  I )
717, 46, 703syl 18 . . . . . . . 8  |-  ( ph  ->  ran  F  =  I )
7271imaeq2d 5168 . . . . . . 7  |-  ( ph  ->  ( S " ran  F )  =  ( S
" I ) )
73 ffn 5728 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
74 fnima 5694 . . . . . . . 8  |-  ( S  Fn  I  ->  ( S " I )  =  ran  S )
7514, 73, 743syl 18 . . . . . . 7  |-  ( ph  ->  ( S " I
)  =  ran  S
)
7672, 75eqtrd 2485 . . . . . 6  |-  ( ph  ->  ( S " ran  F )  =  ran  S
)
7769, 76syl5eq 2497 . . . . 5  |-  ( ph  ->  ran  ( S  o.  F )  =  ran  S )
7877unieqd 4208 . . . 4  |-  ( ph  ->  U. ran  ( S  o.  F )  = 
U. ran  S )
7978fveq2d 5869 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
803dprdspan 17660 . . . 4  |-  ( G dom DProd  ( S  o.  F )  ->  ( G DProd  ( S  o.  F
) )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  ( S  o.  F
) ) )
8168, 80syl 17 . . 3  |-  ( ph  ->  ( G DProd  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  ( S  o.  F )
) )
823dprdspan 17660 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
834, 82syl 17 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
8479, 81, 833eqtr4d 2495 . 2  |-  ( ph  ->  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S
) )
8568, 84jca 535 1  |-  ( ph  ->  ( G dom DProd  ( S  o.  F )  /\  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   {csn 3968   U.cuni 4198   class class class wbr 4402   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    o. ccom 4838   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   0gc0g 15338  mrClscmrc 15489   Grpcgrp 16669  SubGrpcsubg 16811  Cntzccntz 16969   DProd cdprd 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-gim 16923  df-cntz 16971  df-oppg 16997  df-cmn 17432  df-dprd 17627
This theorem is referenced by:  dprdf1  17666  ablfaclem2  17719
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