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Theorem dprdf1o 16519
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 2441 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2441 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2441 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdf1o.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
5 dprdgrp 16479 . . . 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 5637 . . . . 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 16469 . . . . . . 7  |-  Rel  dom DProd
1211brrelex2i 4876 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
13 dmexg 6508 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
144, 12, 133syl 20 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
1510, 14eqeltrrd 2516 . . . 4  |-  ( ph  ->  I  e.  _V )
16 f1dmex 6546 . . . 4  |-  ( ( F : J -1-1-> I  /\  I  e.  _V )  ->  J  e.  _V )
179, 15, 16syl2anc 656 . . 3  |-  ( ph  ->  J  e.  _V )
184, 10dprdf2 16481 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
19 f1of 5638 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J
--> I )
207, 19syl 16 . . . 4  |-  ( ph  ->  F : J --> I )
21 fco 5565 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  F : J --> I )  -> 
( S  o.  F
) : J --> (SubGrp `  G ) )
2218, 20, 21syl2anc 656 . . 3  |-  ( ph  ->  ( S  o.  F
) : J --> (SubGrp `  G ) )
234adantr 462 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  G dom DProd  S )
2410adantr 462 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  dom  S  =  I )
2520adantr 462 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J --> I )
26 simpr1 989 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  e.  J )
27 ffvelrn 5838 . . . . . 6  |-  ( ( F : J --> I  /\  x  e.  J )  ->  ( F `  x
)  e.  I )
2825, 26, 27syl2anc 656 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  e.  I )
29 simpr2 990 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
y  e.  J )
30 ffvelrn 5838 . . . . . 6  |-  ( ( F : J --> I  /\  y  e.  J )  ->  ( F `  y
)  e.  I )
3125, 29, 30syl2anc 656 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  y
)  e.  I )
32 simpr3 991 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  x  =/=  y )
339adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  ->  F : J -1-1-> I )
34 f1fveq 5972 . . . . . . . 8  |-  ( ( F : J -1-1-> I  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
( F `  x
)  =  ( F `
 y )  <->  x  =  y ) )
3533, 26, 29, 34syl12anc 1211 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
3635necon3bid 2641 . . . . . 6  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( F `  x )  =/=  ( F `  y )  <->  x  =/=  y ) )
3732, 36mpbird 232 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( F `  x
)  =/=  ( F `
 y ) )
3823, 24, 28, 31, 37, 1dprdcntz 16482 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( S `  ( F `  x )
)  C_  ( (Cntz `  G ) `  ( S `  ( F `  y ) ) ) )
39 fvco3 5765 . . . . 5  |-  ( ( F : J --> I  /\  x  e.  J )  ->  ( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
4025, 26, 39syl2anc 656 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  =  ( S `
 ( F `  x ) ) )
41 fvco3 5765 . . . . . 6  |-  ( ( F : J --> I  /\  y  e.  J )  ->  ( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
4225, 29, 41syl2anc 656 . . . . 5  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  y
)  =  ( S `
 ( F `  y ) ) )
4342fveq2d 5692 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( ( S  o.  F ) `  y ) )  =  ( (Cntz `  G
) `  ( S `  ( F `  y
) ) ) )
4438, 40, 433sstr4d 3396 . . 3  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( ( S  o.  F ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  o.  F
) `  y )
) )
4520, 39sylan 468 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) `  x )  =  ( S `  ( F `  x ) ) )
46 imaco 5340 . . . . . . . . 9  |-  ( ( S  o.  F )
" ( J  \  { x } ) )  =  ( S
" ( F "
( J  \  {
x } ) ) )
477adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  F : J -1-1-onto-> I )
48 dff1o3 5644 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  <->  ( F : J -onto-> I  /\  Fun  `' F ) )
4948simprbi 461 . . . . . . . . . . . 12  |-  ( F : J -1-1-onto-> I  ->  Fun  `' F )
50 imadif 5490 . . . . . . . . . . . 12  |-  ( Fun  `' F  ->  ( F
" ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
5147, 49, 503syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( ( F " J ) 
\  ( F " { x } ) ) )
52 f1ofo 5645 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  ->  F : J -onto-> I )
53 foima 5622 . . . . . . . . . . . . 13  |-  ( F : J -onto-> I  -> 
( F " J
)  =  I )
5447, 52, 533syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " J )  =  I )
55 f1ofn 5639 . . . . . . . . . . . . . . 15  |-  ( F : J -1-1-onto-> I  ->  F  Fn  J )
567, 55syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  J )
57 fnsnfv 5748 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  J  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F " { x } ) )
5856, 57sylan 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F
" { x }
) )
5958eqcomd 2446 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " { x }
)  =  { ( F `  x ) } )
6054, 59difeq12d 3472 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " J
)  \  ( F " { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6151, 60eqtrd 2473 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6261imaeq2d 5166 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( S " ( F "
( J  \  {
x } ) ) )  =  ( S
" ( I  \  { ( F `  x ) } ) ) )
6346, 62syl5eq 2485 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) " ( J 
\  { x }
) )  =  ( S " ( I 
\  { ( F `
 x ) } ) ) )
6463unieqd 4098 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  U. (
( S  o.  F
) " ( J 
\  { x }
) )  =  U. ( S " ( I 
\  { ( F `
 x ) } ) ) )
6564fveq2d 5692 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  o.  F ) " ( J  \  { x }
) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { ( F `  x ) } ) ) ) )
6645, 65ineq12d 3550 . . . . 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 462 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  G dom DProd  S )
6810adantr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  dom  S  =  I )
6920, 27sylan 468 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F `  x )  e.  I )
7067, 68, 69, 2, 3dprddisj 16483 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( S `  ( F `  x )
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) )  =  { ( 0g `  G ) } )
7166, 70eqtrd 2473 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
72 eqimss 3405 . . . 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 16471 . 2  |-  ( ph  ->  G dom DProd  ( S  o.  F ) )
75 rnco2 5342 . . . . . 6  |-  ran  ( S  o.  F )  =  ( S " ran  F )
76 forn 5620 . . . . . . . . 9  |-  ( F : J -onto-> I  ->  ran  F  =  I )
777, 52, 763syl 20 . . . . . . . 8  |-  ( ph  ->  ran  F  =  I )
7877imaeq2d 5166 . . . . . . 7  |-  ( ph  ->  ( S " ran  F )  =  ( S
" I ) )
79 ffn 5556 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
80 fnima 5526 . . . . . . . 8  |-  ( S  Fn  I  ->  ( S " I )  =  ran  S )
8118, 79, 803syl 20 . . . . . . 7  |-  ( ph  ->  ( S " I
)  =  ran  S
)
8278, 81eqtrd 2473 . . . . . 6  |-  ( ph  ->  ( S " ran  F )  =  ran  S
)
8375, 82syl5eq 2485 . . . . 5  |-  ( ph  ->  ran  ( S  o.  F )  =  ran  S )
8483unieqd 4098 . . . 4  |-  ( ph  ->  U. ran  ( S  o.  F )  = 
U. ran  S )
8584fveq2d 5692 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
863dprdspan 16514 . . . 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 16514 . . . 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 2483 . 2  |-  ( ph  ->  ( G DProd  ( S  o.  F ) )  =  ( G DProd  S
) )
9174, 90jca 529 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 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    \ cdif 3322    i^i cin 3324    C_ wss 3325   {csn 3874   U.cuni 4088   class class class wbr 4289   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839    o. ccom 4840   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   0gc0g 14374  mrClscmrc 14517   Grpcgrp 15406  SubGrpcsubg 15668  Cntzccntz 15826   DProd cdprd 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-gim 15780  df-cntz 15828  df-oppg 15854  df-cmn 16272  df-dprd 16467
This theorem is referenced by:  dprdf1  16520  ablfaclem2  16577
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