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Theorem dprdf1o 15517
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 2387 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2387 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2387 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdf1o.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
5 dprdgrp 15490 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
64, 5syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdf1o.3 . . . . 5  |-  ( ph  ->  F : J -1-1-onto-> I )
8 f1of1 5613 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J -1-1-> I )
97, 8syl 16 . . . 4  |-  ( ph  ->  F : J -1-1-> I
)
10 dprdf1o.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
11 reldmdprd 15485 . . . . . . 7  |-  Rel  dom DProd
1211brrelex2i 4859 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
13 dmexg 5070 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
144, 12, 133syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
1510, 14eqeltrrd 2462 . . . 4  |-  ( ph  ->  I  e.  _V )
16 f1dmex 5910 . . . 4  |-  ( ( F : J -1-1-> I  /\  I  e.  _V )  ->  J  e.  _V )
179, 15, 16syl2anc 643 . . 3  |-  ( ph  ->  J  e.  _V )
184, 10dprdf2 15492 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
19 f1of 5614 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J
--> I )
207, 19syl 16 . . . 4  |-  ( ph  ->  F : J --> I )
21 fco 5540 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  F : J --> I )  -> 
( S  o.  F
) : J --> (SubGrp `  G ) )
2218, 20, 21syl2anc 643 . . 3  |-  ( ph  ->  ( S  o.  F
) : J --> (SubGrp `  G ) )
234adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  G dom DProd  S )
2410adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  dom  S  =  I )
2520adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J --> I )
26 simpr1 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  e.  J )
27 ffvelrn 5807 . . . . . 6  |-  ( ( F : J --> I  /\  x  e.  J )  ->  ( F `  x
)  e.  I )
2825, 26, 27syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  e.  I )
29 simpr2 964 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
y  e.  J )
30 ffvelrn 5807 . . . . . 6  |-  ( ( F : J --> I  /\  y  e.  J )  ->  ( F `  y
)  e.  I )
3125, 29, 30syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  y
)  e.  I )
32 simpr3 965 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  =/=  y )
339adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J -1-1-> I )
34 f1fveq 5947 . . . . . . . 8  |-  ( ( F : J -1-1-> I  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
( F `  x
)  =  ( F `
 y )  <->  x  =  y ) )
3533, 26, 29, 34syl12anc 1182 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
3635necon3bid 2585 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =/=  ( F `  y )  <->  x  =/=  y ) )
3732, 36mpbird 224 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  =/=  ( F `
 y ) )
3823, 24, 28, 31, 37, 1dprdcntz 15493 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( S `  ( F `  x )
)  C_  ( (Cntz `  G ) `  ( S `  ( F `  y ) ) ) )
39 fvco3 5739 . . . . 5  |-  ( ( F : J --> I  /\  x  e.  J )  ->  ( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
4025, 26, 39syl2anc 643 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
41 fvco3 5739 . . . . . 6  |-  ( ( F : J --> I  /\  y  e.  J )  ->  ( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
4225, 29, 41syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
4342fveq2d 5672 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( ( S  o.  F ) `  y ) )  =  ( (Cntz `  G
) `  ( S `  ( F `  y
) ) ) )
4438, 40, 433sstr4d 3334 . . 3  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  o.  F
) `  y )
) )
4520, 39sylan 458 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) `  x )  =  ( S `  ( F `  x ) ) )
46 imaco 5315 . . . . . . . . 9  |-  ( ( S  o.  F )
" ( J  \  { x } ) )  =  ( S
" ( F "
( J  \  {
x } ) ) )
477adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  F : J -1-1-onto-> I )
48 dff1o3 5620 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  <->  ( F : J -onto-> I  /\  Fun  `' F ) )
4948simprbi 451 . . . . . . . . . . . 12  |-  ( F : J -1-1-onto-> I  ->  Fun  `' F )
50 imadif 5468 . . . . . . . . . . . 12  |-  ( Fun  `' F  ->  ( F
" ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
5147, 49, 503syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
52 f1ofo 5621 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  ->  F : J -onto-> I )
53 foima 5598 . . . . . . . . . . . . 13  |-  ( F : J -onto-> I  -> 
( F " J
)  =  I )
5447, 52, 533syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " J )  =  I )
55 f1ofn 5615 . . . . . . . . . . . . . . 15  |-  ( F : J -1-1-onto-> I  ->  F  Fn  J )
567, 55syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  J )
57 fnsnfv 5725 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  J  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F " { x } ) )
5856, 57sylan 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F
" { x }
) )
5958eqcomd 2392 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " { x }
)  =  { ( F `  x ) } )
6054, 59difeq12d 3409 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " J
)  \  ( F " { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6151, 60eqtrd 2419 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6261imaeq2d 5143 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( S " ( F "
( J  \  {
x } ) ) )  =  ( S
" ( I  \  { ( F `  x ) } ) ) )
6346, 62syl5eq 2431 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) " ( J 
\  { x }
) )  =  ( S " ( I 
\  { ( F `
 x ) } ) ) )
6463unieqd 3968 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  U. (
( S  o.  F
) " ( J 
\  { x }
) )  =  U. ( S " ( I 
\  { ( F `
 x ) } ) ) )
6564fveq2d 5672 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  o.  F ) " ( J  \  { x }
) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { ( F `  x ) } ) ) ) )
6645, 65ineq12d 3486 . . . . 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 ) } ) ) ) ) )
674adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  G dom DProd  S )
6810adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  dom  S  =  I )
6920, 27sylan 458 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F `  x )  e.  I )
7067, 68, 69, 2, 3dprddisj 15494 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( S `  ( F `  x )
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) )  =  { ( 0g `  G ) } )
7166, 70eqtrd 2419 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
72 eqimss 3343 . . . 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
) } )
7371, 72syl 16 . . 3  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
741, 2, 3, 6, 17, 22, 44, 73dmdprdd 15487 . 2  |-  ( ph  ->  G dom DProd  ( S  o.  F ) )
75 rnco2 5317 . . . . . 6  |-  ran  ( S  o.  F )  =  ( S " ran  F )
76 forn 5596 . . . . . . . . 9  |-  ( F : J -onto-> I  ->  ran  F  =  I )
777, 52, 763syl 19 . . . . . . . 8  |-  ( ph  ->  ran  F  =  I )
7877imaeq2d 5143 . . . . . . 7  |-  ( ph  ->  ( S " ran  F )  =  ( S
" I ) )
79 ffn 5531 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
80 fnima 5503 . . . . . . . 8  |-  ( S  Fn  I  ->  ( S " I )  =  ran  S )
8118, 79, 803syl 19 . . . . . . 7  |-  ( ph  ->  ( S " I
)  =  ran  S
)
8278, 81eqtrd 2419 . . . . . 6  |-  ( ph  ->  ( S " ran  F )  =  ran  S
)
8375, 82syl5eq 2431 . . . . 5  |-  ( ph  ->  ran  ( S  o.  F )  =  ran  S )
8483unieqd 3968 . . . 4  |-  ( ph  ->  U. ran  ( S  o.  F )  = 
U. ran  S )
8584fveq2d 5672 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
863dprdspan 15512 . . . 4  |-  ( G dom DProd  ( S  o.  F )  ->  ( G DProd  ( S  o.  F
) )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  ( S  o.  F
) ) )
8774, 86syl 16 . . 3  |-  ( ph  ->  ( G DProd  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  ( S  o.  F )
) )
883dprdspan 15512 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
894, 88syl 16 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
9085, 87, 893eqtr4d 2429 . 2  |-  ( ph  ->  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S
) )
9174, 90jca 519 1  |-  ( ph  ->  ( G dom DProd  ( S  o.  F )  /\  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    \ cdif 3260    i^i cin 3262    C_ wss 3263   {csn 3757   U.cuni 3957   class class class wbr 4153   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821    o. ccom 4822   Fun wfun 5388    Fn wfn 5389   -->wf 5390   -1-1->wf1 5391   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020   0gc0g 13650  mrClscmrc 13735   Grpcgrp 14612  SubGrpcsubg 14865  Cntzccntz 15041   DProd cdprd 15481
This theorem is referenced by:  dprdf1  15518  ablfaclem2  15571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-gim 14973  df-cntz 15043  df-oppg 15069  df-cmn 15341  df-dprd 15483
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