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Theorem gicsubgen 14758
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 14749 . . 3  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3477 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. a  a  e.  ( R GrpIso  S ) )
31, 2bitri 240 . 2  |-  ( R 
~=ph𝑔  S 
<->  E. a  a  e.  ( R GrpIso  S ) )
4 fvex 5555 . . . . 5  |-  (SubGrp `  R )  e.  _V
54a1i 10 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  e.  _V )
6 fvex 5555 . . . . 5  |-  (SubGrp `  S )  e.  _V
76a1i 10 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  S
)  e.  _V )
8 vex 2804 . . . . . 6  |-  a  e. 
_V
9 imaexg 5042 . . . . . 6  |-  ( a  e.  _V  ->  (
a " b )  e.  _V )
108, 9ax-mp 8 . . . . 5  |-  ( a
" b )  e. 
_V
1110a1ii 24 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( b  e.  (SubGrp `  R )  ->  ( a " b
)  e.  _V )
)
128cnvex 5225 . . . . . 6  |-  `' a  e.  _V
13 imaexg 5042 . . . . . 6  |-  ( `' a  e.  _V  ->  ( `' a " c
)  e.  _V )
1412, 13ax-mp 8 . . . . 5  |-  ( `' a " c )  e.  _V
1514a1ii 24 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( c  e.  (SubGrp `  S )  ->  ( `' a "
c )  e.  _V ) )
16 gimghm 14744 . . . . . . . . 9  |-  ( a  e.  ( R GrpIso  S
)  ->  a  e.  ( R  GrpHom  S ) )
17 ghmima 14719 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
1816, 17sylan 457 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
19 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
2119, 20gimf1o 14743 . . . . . . . . . . 11  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
22 f1of1 5487 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -1-1-> ( Base `  S ) )
2321, 22syl 15 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-> ( Base `  S
) )
2419subgss 14638 . . . . . . . . . 10  |-  ( b  e.  (SubGrp `  R
)  ->  b  C_  ( Base `  R )
)
25 f1imacnv 5505 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -1-1-> ( Base `  S )  /\  b  C_  ( Base `  R
) )  ->  ( `' a " (
a " b ) )  =  b )
2623, 24, 25syl2an 463 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( `' a " ( a "
b ) )  =  b )
2726eqcomd 2301 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  b  =  ( `' a " (
a " b ) ) )
2818, 27jca 518 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( (
a " b )  e.  (SubGrp `  S
)  /\  b  =  ( `' a " (
a " b ) ) ) )
29 eleq1 2356 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  <->  ( a " b )  e.  (SubGrp `  S )
) )
30 imaeq2 5024 . . . . . . . . 9  |-  ( c  =  ( a "
b )  ->  ( `' a " c
)  =  ( `' a " ( a
" b ) ) )
3130eqeq2d 2307 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
b  =  ( `' a " c )  <-> 
b  =  ( `' a " ( a
" b ) ) ) )
3229, 31anbi12d 691 . . . . . . 7  |-  ( c  =  ( a "
b )  ->  (
( c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) )  <-> 
( ( a "
b )  e.  (SubGrp `  S )  /\  b  =  ( `' a
" ( a "
b ) ) ) ) )
3328, 32syl5ibrcom 213 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) ) )
3433impr 602 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) )  ->  ( c  e.  (SubGrp `  S )  /\  b  =  ( `' a " c
) ) )
35 ghmpreima 14720 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
3616, 35sylan 457 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
37 f1ofo 5495 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -onto-> ( Base `  S ) )
3821, 37syl 15 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -onto->
( Base `  S )
)
3920subgss 14638 . . . . . . . . . 10  |-  ( c  e.  (SubGrp `  S
)  ->  c  C_  ( Base `  S )
)
40 foimacnv 5506 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -onto-> ( Base `  S )  /\  c  C_  ( Base `  S
) )  ->  (
a " ( `' a " c ) )  =  c )
4138, 39, 40syl2an 463 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( a " ( `' a
" c ) )  =  c )
4241eqcomd 2301 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  c  =  ( a " ( `' a " c
) ) )
4336, 42jca 518 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( ( `' a " c
)  e.  (SubGrp `  R )  /\  c  =  ( a "
( `' a "
c ) ) ) )
44 eleq1 2356 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  <->  ( `' a " c )  e.  (SubGrp `  R )
) )
45 imaeq2 5024 . . . . . . . . 9  |-  ( b  =  ( `' a
" c )  -> 
( a " b
)  =  ( a
" ( `' a
" c ) ) )
4645eqeq2d 2307 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( c  =  ( a " b )  <-> 
c  =  ( a
" ( `' a
" c ) ) ) )
4744, 46anbi12d 691 . . . . . . 7  |-  ( b  =  ( `' a
" c )  -> 
( ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) )  <->  ( ( `' a " c )  e.  (SubGrp `  R
)  /\  c  =  ( a " ( `' a " c
) ) ) ) )
4843, 47syl5ibrcom 213 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) ) )
4948impr 602 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) )  ->  ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) ) )
5034, 49impbida 805 . . . 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 6913 . . 3  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  ~~  (SubGrp `  S
) )
5251exlimiv 1624 . 2  |-  ( E. a  a  e.  ( R GrpIso  S )  -> 
(SubGrp `  R )  ~~  (SubGrp `  S )
)
533, 52sylbi 187 1  |-  ( R 
~=ph𝑔  S  ->  (SubGrp `  R )  ~~  (SubGrp `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039   `'ccnv 4704   "cima 4708   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ~~ cen 6876   Basecbs 13164  SubGrpcsubg 14631    GrpHom cghm 14696   GrpIso cgim 14737    ~=ph𝑔 cgic 14738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-ghm 14697  df-gim 14739  df-gic 14740
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