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Theorem gicsubgen 15811
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 15802 . . 3  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3651 . . 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 5706 . . . . 5  |-  (SubGrp `  R )  e.  _V
54a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  e.  _V )
6 fvex 5706 . . . . 5  |-  (SubGrp `  S )  e.  _V
76a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  S
)  e.  _V )
8 vex 2980 . . . . . 6  |-  a  e. 
_V
9 imaexg 6520 . . . . . 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 6530 . . . . . 6  |-  `' a  e.  _V
13 imaexg 6520 . . . . . 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 15797 . . . . . . . . 9  |-  ( a  e.  ( R GrpIso  S
)  ->  a  e.  ( R  GrpHom  S ) )
17 ghmima 15772 . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
2119, 20gimf1o 15796 . . . . . . . . . . 11  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
22 f1of1 5645 . . . . . . . . . . 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 15687 . . . . . . . . . 10  |-  ( b  e.  (SubGrp `  R
)  ->  b  C_  ( Base `  R )
)
25 f1imacnv 5662 . . . . . . . . . 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 2448 . . . . . . . 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 2503 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  <->  ( a " b )  e.  (SubGrp `  S )
) )
30 imaeq2 5170 . . . . . . . . 9  |-  ( c  =  ( a "
b )  ->  ( `' a " c
)  =  ( `' a " ( a
" b ) ) )
3130eqeq2d 2454 . . . . . . . 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 15773 . . . . . . . . 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 5653 . . . . . . . . . . 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 15687 . . . . . . . . . 10  |-  ( c  e.  (SubGrp `  S
)  ->  c  C_  ( Base `  S )
)
40 foimacnv 5663 . . . . . . . . . 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 2448 . . . . . . . 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 2503 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  <->  ( `' a " c )  e.  (SubGrp `  R )
) )
45 imaeq2 5170 . . . . . . . . 9  |-  ( b  =  ( `' a
" c )  -> 
( a " b
)  =  ( a
" ( `' a
" c ) ) )
4645eqeq2d 2454 . . . . . . . 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 828 . . . 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 7350 . . 3  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  ~~  (SubGrp `  S
) )
5251exlimiv 1688 . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   _Vcvv 2977    C_ wss 3333   (/)c0 3642   class class class wbr 4297   `'ccnv 4844   "cima 4848   -1-1->wf1 5420   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    ~~ cen 7312   Basecbs 14179  SubGrpcsubg 15680    GrpHom cghm 15749   GrpIso cgim 15790    ~=ph𝑔 cgic 15791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-subg 15683  df-ghm 15750  df-gim 15792  df-gic 15793
This theorem is referenced by: (None)
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