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Theorem gicsubgen 16131
Description: A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
gicsubgen  |-  ( R 
~=ph𝑔  S  ->  (SubGrp `  R )  ~~  (SubGrp `  S )
)

Proof of Theorem gicsubgen
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brgic 16122 . . 3  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3794 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. a  a  e.  ( R GrpIso  S ) )
31, 2bitri 249 . 2  |-  ( R 
~=ph𝑔  S 
<->  E. a  a  e.  ( R GrpIso  S ) )
4 fvex 5876 . . . . 5  |-  (SubGrp `  R )  e.  _V
54a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  e.  _V )
6 fvex 5876 . . . . 5  |-  (SubGrp `  S )  e.  _V
76a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  S
)  e.  _V )
8 vex 3116 . . . . . 6  |-  a  e. 
_V
9 imaexg 6721 . . . . . 6  |-  ( a  e.  _V  ->  (
a " b )  e.  _V )
108, 9ax-mp 5 . . . . 5  |-  ( a
" b )  e. 
_V
1110a1ii 27 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( b  e.  (SubGrp `  R )  ->  ( a " b
)  e.  _V )
)
128cnvex 6731 . . . . . 6  |-  `' a  e.  _V
13 imaexg 6721 . . . . . 6  |-  ( `' a  e.  _V  ->  ( `' a " c
)  e.  _V )
1412, 13ax-mp 5 . . . . 5  |-  ( `' a " c )  e.  _V
1514a1ii 27 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( c  e.  (SubGrp `  S )  ->  ( `' a "
c )  e.  _V ) )
16 gimghm 16117 . . . . . . . . 9  |-  ( a  e.  ( R GrpIso  S
)  ->  a  e.  ( R  GrpHom  S ) )
17 ghmima 16092 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
1816, 17sylan 471 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
19 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
2119, 20gimf1o 16116 . . . . . . . . . . 11  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
22 f1of1 5815 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -1-1-> ( Base `  S ) )
2321, 22syl 16 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-> ( Base `  S
) )
2419subgss 16007 . . . . . . . . . 10  |-  ( b  e.  (SubGrp `  R
)  ->  b  C_  ( Base `  R )
)
25 f1imacnv 5832 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -1-1-> ( Base `  S )  /\  b  C_  ( Base `  R
) )  ->  ( `' a " (
a " b ) )  =  b )
2623, 24, 25syl2an 477 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( `' a " ( a "
b ) )  =  b )
2726eqcomd 2475 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  b  =  ( `' a " (
a " b ) ) )
2818, 27jca 532 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( (
a " b )  e.  (SubGrp `  S
)  /\  b  =  ( `' a " (
a " b ) ) ) )
29 eleq1 2539 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  <->  ( a " b )  e.  (SubGrp `  S )
) )
30 imaeq2 5333 . . . . . . . . 9  |-  ( c  =  ( a "
b )  ->  ( `' a " c
)  =  ( `' a " ( a
" b ) ) )
3130eqeq2d 2481 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
b  =  ( `' a " c )  <-> 
b  =  ( `' a " ( a
" b ) ) ) )
3229, 31anbi12d 710 . . . . . . 7  |-  ( c  =  ( a "
b )  ->  (
( c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) )  <-> 
( ( a "
b )  e.  (SubGrp `  S )  /\  b  =  ( `' a
" ( a "
b ) ) ) ) )
3328, 32syl5ibrcom 222 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) ) )
3433impr 619 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) )  ->  ( c  e.  (SubGrp `  S )  /\  b  =  ( `' a " c
) ) )
35 ghmpreima 16093 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
3616, 35sylan 471 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
37 f1ofo 5823 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -onto-> ( Base `  S ) )
3821, 37syl 16 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -onto->
( Base `  S )
)
3920subgss 16007 . . . . . . . . . 10  |-  ( c  e.  (SubGrp `  S
)  ->  c  C_  ( Base `  S )
)
40 foimacnv 5833 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -onto-> ( Base `  S )  /\  c  C_  ( Base `  S
) )  ->  (
a " ( `' a " c ) )  =  c )
4138, 39, 40syl2an 477 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( a " ( `' a
" c ) )  =  c )
4241eqcomd 2475 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  c  =  ( a " ( `' a " c
) ) )
4336, 42jca 532 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( ( `' a " c
)  e.  (SubGrp `  R )  /\  c  =  ( a "
( `' a "
c ) ) ) )
44 eleq1 2539 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  <->  ( `' a " c )  e.  (SubGrp `  R )
) )
45 imaeq2 5333 . . . . . . . . 9  |-  ( b  =  ( `' a
" c )  -> 
( a " b
)  =  ( a
" ( `' a
" c ) ) )
4645eqeq2d 2481 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( c  =  ( a " b )  <-> 
c  =  ( a
" ( `' a
" c ) ) ) )
4744, 46anbi12d 710 . . . . . . 7  |-  ( b  =  ( `' a
" c )  -> 
( ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) )  <->  ( ( `' a " c )  e.  (SubGrp `  R
)  /\  c  =  ( a " ( `' a " c
) ) ) ) )
4843, 47syl5ibrcom 222 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) ) )
4948impr 619 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) )  ->  ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) ) )
5034, 49impbida 830 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( (
b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) )  <->  ( c  e.  (SubGrp `  S )  /\  b  =  ( `' a " c
) ) ) )
515, 7, 11, 15, 50en2d 7551 . . 3  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  ~~  (SubGrp `  S
) )
5251exlimiv 1698 . 2  |-  ( E. a  a  e.  ( R GrpIso  S )  -> 
(SubGrp `  R )  ~~  (SubGrp `  S )
)
533, 52sylbi 195 1  |-  ( R 
~=ph𝑔  S  ->  (SubGrp `  R )  ~~  (SubGrp `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447   `'ccnv 4998   "cima 5002   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    ~~ cen 7513   Basecbs 14490  SubGrpcsubg 16000    GrpHom cghm 16069   GrpIso cgim 16110    ~=ph𝑔 cgic 16111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-subg 16003  df-ghm 16070  df-gim 16112  df-gic 16113
This theorem is referenced by: (None)
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