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Theorem dprdf1o 15553
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 2412 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2412 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2412 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdf1o.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
5 dprdgrp 15526 . . . 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 5640 . . . . 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 15521 . . . . . . 7  |-  Rel  dom DProd
1211brrelex2i 4886 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
13 dmexg 5097 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
144, 12, 133syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
1510, 14eqeltrrd 2487 . . . 4  |-  ( ph  ->  I  e.  _V )
16 f1dmex 5938 . . . 4  |-  ( ( F : J -1-1-> I  /\  I  e.  _V )  ->  J  e.  _V )
179, 15, 16syl2anc 643 . . 3  |-  ( ph  ->  J  e.  _V )
184, 10dprdf2 15528 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
19 f1of 5641 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J
--> I )
207, 19syl 16 . . . 4  |-  ( ph  ->  F : J --> I )
21 fco 5567 . . . 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 5835 . . . . . 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 5835 . . . . . 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 5975 . . . . . . . 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 2610 . . . . . 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 15529 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( S `  ( F `  x )
)  C_  ( (Cntz `  G ) `  ( S `  ( F `  y ) ) ) )
39 fvco3 5767 . . . . 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 5767 . . . . . 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 5699 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( ( S  o.  F ) `  y ) )  =  ( (Cntz `  G
) `  ( S `  ( F `  y
) ) ) )
4438, 40, 433sstr4d 3359 . . 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 5342 . . . . . . . . 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 5647 . . . . . . . . . . . . 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 5495 . . . . . . . . . . . 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 5648 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  ->  F : J -onto-> I )
53 foima 5625 . . . . . . . . . . . . 13  |-  ( F : J -onto-> I  -> 
( F " J
)  =  I )
5447, 52, 533syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " J )  =  I )
55 f1ofn 5642 . . . . . . . . . . . . . . 15  |-  ( F : J -1-1-onto-> I  ->  F  Fn  J )
567, 55syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  J )
57 fnsnfv 5753 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  J  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F " { x } ) )
5856, 57sylan 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F
" { x }
) )
5958eqcomd 2417 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " { x }
)  =  { ( F `  x ) } )
6054, 59difeq12d 3434 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " J
)  \  ( F " { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6151, 60eqtrd 2444 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6261imaeq2d 5170 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( S " ( F "
( J  \  {
x } ) ) )  =  ( S
" ( I  \  { ( F `  x ) } ) ) )
6346, 62syl5eq 2456 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) " ( J 
\  { x }
) )  =  ( S " ( I 
\  { ( F `
 x ) } ) ) )
6463unieqd 3994 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  U. (
( S  o.  F
) " ( J 
\  { x }
) )  =  U. ( S " ( I 
\  { ( F `
 x ) } ) ) )
6564fveq2d 5699 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  o.  F ) " ( J  \  { x }
) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { ( F `  x ) } ) ) ) )
6645, 65ineq12d 3511 . . . . 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 15530 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( S `  ( F `  x )
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) )  =  { ( 0g `  G ) } )
7166, 70eqtrd 2444 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
72 eqimss 3368 . . . 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 15523 . 2  |-  ( ph  ->  G dom DProd  ( S  o.  F ) )
75 rnco2 5344 . . . . . 6  |-  ran  ( S  o.  F )  =  ( S " ran  F )
76 forn 5623 . . . . . . . . 9  |-  ( F : J -onto-> I  ->  ran  F  =  I )
777, 52, 763syl 19 . . . . . . . 8  |-  ( ph  ->  ran  F  =  I )
7877imaeq2d 5170 . . . . . . 7  |-  ( ph  ->  ( S " ran  F )  =  ( S
" I ) )
79 ffn 5558 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
80 fnima 5530 . . . . . . . 8  |-  ( S  Fn  I  ->  ( S " I )  =  ran  S )
8118, 79, 803syl 19 . . . . . . 7  |-  ( ph  ->  ( S " I
)  =  ran  S
)
8278, 81eqtrd 2444 . . . . . 6  |-  ( ph  ->  ( S " ran  F )  =  ran  S
)
8375, 82syl5eq 2456 . . . . 5  |-  ( ph  ->  ran  ( S  o.  F )  =  ran  S )
8483unieqd 3994 . . . 4  |-  ( ph  ->  U. ran  ( S  o.  F )  = 
U. ran  S )
8584fveq2d 5699 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
863dprdspan 15548 . . . 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 15548 . . . 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 2454 . 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 1721    =/= wne 2575   _Vcvv 2924    \ cdif 3285    i^i cin 3287    C_ wss 3288   {csn 3782   U.cuni 3983   class class class wbr 4180   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848    o. ccom 4849   Fun wfun 5415    Fn wfn 5416   -->wf 5417   -1-1->wf1 5418   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   0gc0g 13686  mrClscmrc 13771   Grpcgrp 14648  SubGrpcsubg 14901  Cntzccntz 15077   DProd cdprd 15517
This theorem is referenced by:  dprdf1  15554  ablfaclem2  15607
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-ghm 14967  df-gim 15009  df-cntz 15079  df-oppg 15105  df-cmn 15377  df-dprd 15519
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