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Theorem dprdcntz2 15273
Description: The function  S is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz2.1  |-  ( ph  ->  G dom DProd  S )
dprdcntz2.2  |-  ( ph  ->  dom  S  =  I )
dprdcntz2.c  |-  ( ph  ->  C  C_  I )
dprdcntz2.d  |-  ( ph  ->  D  C_  I )
dprdcntz2.i  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
dprdcntz2.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
dprdcntz2  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )

Proof of Theorem dprdcntz2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdcntz2.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
2 dprdcntz2.2 . . . 4  |-  ( ph  ->  dom  S  =  I )
3 dprdcntz2.c . . . 4  |-  ( ph  ->  C  C_  I )
41, 2, 3dprdres 15263 . . 3  |-  ( ph  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) ) )
54simpld 445 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
6 dmres 4976 . . 3  |-  dom  ( S  |`  C )  =  ( C  i^i  dom  S )
73, 2sseqtr4d 3215 . . . 4  |-  ( ph  ->  C  C_  dom  S )
8 df-ss 3166 . . . 4  |-  ( C 
C_  dom  S  <->  ( C  i^i  dom  S )  =  C )
97, 8sylib 188 . . 3  |-  ( ph  ->  ( C  i^i  dom  S )  =  C )
106, 9syl5eq 2327 . 2  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
11 dprdgrp 15240 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
121, 11syl 15 . . 3  |-  ( ph  ->  G  e.  Grp )
13 eqid 2283 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
1413dprdssv 15251 . . 3  |-  ( G DProd 
( S  |`  D ) )  C_  ( Base `  G )
15 dprdcntz2.z . . . 4  |-  Z  =  (Cntz `  G )
1613, 15cntzsubg 14812 . . 3  |-  ( ( G  e.  Grp  /\  ( G DProd  ( S  |`  D ) )  C_  ( Base `  G )
)  ->  ( Z `  ( G DProd  ( S  |`  D ) ) )  e.  (SubGrp `  G
) )
1712, 14, 16sylancl 643 . 2  |-  ( ph  ->  ( Z `  ( G DProd  ( S  |`  D ) ) )  e.  (SubGrp `  G ) )
18 fvres 5542 . . . 4  |-  ( x  e.  C  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
1918adantl 452 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
20 dprdcntz2.d . . . . . . . 8  |-  ( ph  ->  D  C_  I )
211, 2, 20dprdres 15263 . . . . . . 7  |-  ( ph  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) ) )
2221simpld 445 . . . . . 6  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
2322adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  G dom DProd  ( S  |`  D ) )
24 dprdsubg 15259 . . . . 5  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
2523, 24syl 15 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
263sselda 3180 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  I )
271, 2dprdf2 15242 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
28 ffvelrn 5663 . . . . . 6  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
2927, 28sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
3026, 29syldan 456 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  e.  (SubGrp `  G )
)
31 dmres 4976 . . . . . . 7  |-  dom  ( S  |`  D )  =  ( D  i^i  dom  S )
3220, 2sseqtr4d 3215 . . . . . . . 8  |-  ( ph  ->  D  C_  dom  S )
33 df-ss 3166 . . . . . . . 8  |-  ( D 
C_  dom  S  <->  ( D  i^i  dom  S )  =  D )
3432, 33sylib 188 . . . . . . 7  |-  ( ph  ->  ( D  i^i  dom  S )  =  D )
3531, 34syl5eq 2327 . . . . . 6  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
3635adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  dom  ( S  |`  D )  =  D )
3712adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  G  e.  Grp )
3813subgss 14622 . . . . . . 7  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  ( Base `  G ) )
3930, 38syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( Base `  G
) )
4013, 15cntzsubg 14812 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( S `  x ) 
C_  ( Base `  G
) )  ->  ( Z `  ( S `  x ) )  e.  (SubGrp `  G )
)
4137, 39, 40syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( Z `  ( S `  x ) )  e.  (SubGrp `  G )
)
42 fvres 5542 . . . . . . 7  |-  ( y  e.  D  ->  (
( S  |`  D ) `
 y )  =  ( S `  y
) )
4342adantl 452 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  (
( S  |`  D ) `
 y )  =  ( S `  y
) )
441ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  G dom DProd  S )
452ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  dom  S  =  I )
4620adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  D  C_  I )
4746sselda 3180 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  y  e.  I )
4826adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  x  e.  I )
49 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  y  e.  D )
50 noel 3459 . . . . . . . . . . . 12  |-  -.  x  e.  (/)
51 elin 3358 . . . . . . . . . . . . 13  |-  ( x  e.  ( C  i^i  D )  <->  ( x  e.  C  /\  x  e.  D ) )
52 dprdcntz2.i . . . . . . . . . . . . . 14  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
5352eleq2d 2350 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( C  i^i  D )  <-> 
x  e.  (/) ) )
5451, 53syl5bbr 250 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  e.  C  /\  x  e.  D )  <->  x  e.  (/) ) )
5550, 54mtbiri 294 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( x  e.  C  /\  x  e.  D ) )
56 imnan 411 . . . . . . . . . . 11  |-  ( ( x  e.  C  ->  -.  x  e.  D
)  <->  -.  ( x  e.  C  /\  x  e.  D ) )
5755, 56sylibr 203 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  C  ->  -.  x  e.  D
) )
5857imp 418 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  -.  x  e.  D )
5958adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  -.  x  e.  D )
60 nelne2 2536 . . . . . . . 8  |-  ( ( y  e.  D  /\  -.  x  e.  D
)  ->  y  =/=  x )
6149, 59, 60syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  y  =/=  x )
6244, 45, 47, 48, 61, 15dprdcntz 15243 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  ( S `  y )  C_  ( Z `  ( S `  x )
) )
6343, 62eqsstrd 3212 . . . . 5  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  D )  ->  (
( S  |`  D ) `
 y )  C_  ( Z `  ( S `
 x ) ) )
6423, 36, 41, 63dprdlub 15261 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( G DProd  ( S  |`  D ) )  C_  ( Z `  ( S `  x
) ) )
6515, 25, 30, 64cntzrecd 14987 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )
6619, 65eqsstrd 3212 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
675, 10, 17, 66dprdlub 15261 1  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprd2da  15277  dmdprdsplit  15282  ablfac1eulem  15307  ablfac1eu  15308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-cmn 15091  df-dprd 15233
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