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Theorem dprdf1o 15590
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 2436 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2436 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2436 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdf1o.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
5 dprdgrp 15563 . . . 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 5673 . . . . 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 15558 . . . . . . 7  |-  Rel  dom DProd
1211brrelex2i 4919 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
13 dmexg 5130 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
144, 12, 133syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
1510, 14eqeltrrd 2511 . . . 4  |-  ( ph  ->  I  e.  _V )
16 f1dmex 5971 . . . 4  |-  ( ( F : J -1-1-> I  /\  I  e.  _V )  ->  J  e.  _V )
179, 15, 16syl2anc 643 . . 3  |-  ( ph  ->  J  e.  _V )
184, 10dprdf2 15565 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
19 f1of 5674 . . . . 5  |-  ( F : J -1-1-onto-> I  ->  F : J
--> I )
207, 19syl 16 . . . 4  |-  ( ph  ->  F : J --> I )
21 fco 5600 . . . 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 5868 . . . . . 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 5868 . . . . . 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 6008 . . . . . . . 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 2636 . . . . . 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 15566 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( S `  ( F `  x )
)  C_  ( (Cntz `  G ) `  ( S `  ( F `  y ) ) ) )
39 fvco3 5800 . . . . 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 5800 . . . . . 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 5732 . . . 4  |-  ( (
ph  /\  ( x  e.  J  /\  y  e.  J  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( ( S  o.  F ) `  y ) )  =  ( (Cntz `  G
) `  ( S `  ( F `  y
) ) ) )
4438, 40, 433sstr4d 3391 . . 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 5375 . . . . . . . . 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 5680 . . . . . . . . . . . . 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 5528 . . . . . . . . . . . 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 5681 . . . . . . . . . . . . 13  |-  ( F : J -1-1-onto-> I  ->  F : J -onto-> I )
53 foima 5658 . . . . . . . . . . . . 13  |-  ( F : J -onto-> I  -> 
( F " J
)  =  I )
5447, 52, 533syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " J )  =  I )
55 f1ofn 5675 . . . . . . . . . . . . . . 15  |-  ( F : J -1-1-onto-> I  ->  F  Fn  J )
567, 55syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  J )
57 fnsnfv 5786 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  J  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F " { x } ) )
5856, 57sylan 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  J )  ->  { ( F `  x ) }  =  ( F
" { x }
) )
5958eqcomd 2441 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  J )  ->  ( F " { x }
)  =  { ( F `  x ) } )
6054, 59difeq12d 3466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " J
)  \  ( F " { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6151, 60eqtrd 2468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  J )  ->  ( F " ( J  \  { x } ) )  =  ( I 
\  { ( F `
 x ) } ) )
6261imaeq2d 5203 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  J )  ->  ( S " ( F "
( J  \  {
x } ) ) )  =  ( S
" ( I  \  { ( F `  x ) } ) ) )
6346, 62syl5eq 2480 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  (
( S  o.  F
) " ( J 
\  { x }
) )  =  ( S " ( I 
\  { ( F `
 x ) } ) ) )
6463unieqd 4026 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  U. (
( S  o.  F
) " ( J 
\  { x }
) )  =  U. ( S " ( I 
\  { ( F `
 x ) } ) ) )
6564fveq2d 5732 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  o.  F ) " ( J  \  { x }
) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { ( F `  x ) } ) ) ) )
6645, 65ineq12d 3543 . . . . 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 15567 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  (
( S `  ( F `  x )
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { ( F `  x ) } ) ) ) )  =  { ( 0g `  G ) } )
7166, 70eqtrd 2468 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  (
( ( S  o.  F ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  o.  F )
" ( J  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
72 eqimss 3400 . . . 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 15560 . 2  |-  ( ph  ->  G dom DProd  ( S  o.  F ) )
75 rnco2 5377 . . . . . 6  |-  ran  ( S  o.  F )  =  ( S " ran  F )
76 forn 5656 . . . . . . . . 9  |-  ( F : J -onto-> I  ->  ran  F  =  I )
777, 52, 763syl 19 . . . . . . . 8  |-  ( ph  ->  ran  F  =  I )
7877imaeq2d 5203 . . . . . . 7  |-  ( ph  ->  ( S " ran  F )  =  ( S
" I ) )
79 ffn 5591 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  S  Fn  I )
80 fnima 5563 . . . . . . . 8  |-  ( S  Fn  I  ->  ( S " I )  =  ran  S )
8118, 79, 803syl 19 . . . . . . 7  |-  ( ph  ->  ( S " I
)  =  ran  S
)
8278, 81eqtrd 2468 . . . . . 6  |-  ( ph  ->  ( S " ran  F )  =  ran  S
)
8375, 82syl5eq 2480 . . . . 5  |-  ( ph  ->  ran  ( S  o.  F )  =  ran  S )
8483unieqd 4026 . . . 4  |-  ( ph  ->  U. ran  ( S  o.  F )  = 
U. ran  S )
8584fveq2d 5732 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  o.  F ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  S ) )
863dprdspan 15585 . . . 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 15585 . . . 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 2478 . 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   {csn 3814   U.cuni 4015   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881    o. ccom 4882   Fun wfun 5448    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   0gc0g 13723  mrClscmrc 13808   Grpcgrp 14685  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
This theorem is referenced by:  dprdf1  15591  ablfaclem2  15644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-gim 15046  df-cntz 15116  df-oppg 15142  df-cmn 15414  df-dprd 15556
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