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Theorem gicsubgen 16893
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 
~=g𝑔  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 16884 . . 3  |-  ( R 
~=g𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3777 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. a  a  e.  ( R GrpIso  S ) )
31, 2bitri 252 . 2  |-  ( R 
~=g𝑔  S 
<->  E. a  a  e.  ( R GrpIso  S ) )
4 fvex 5891 . . . . 5  |-  (SubGrp `  R )  e.  _V
54a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  e.  _V )
6 fvex 5891 . . . . 5  |-  (SubGrp `  S )  e.  _V
76a1i 11 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  S
)  e.  _V )
8 vex 3090 . . . . . 6  |-  a  e. 
_V
9 imaexg 6744 . . . . . 6  |-  ( a  e.  _V  ->  (
a " b )  e.  _V )
108, 9ax-mp 5 . . . . 5  |-  ( a
" b )  e. 
_V
11102a1i 12 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( b  e.  (SubGrp `  R )  ->  ( a " b
)  e.  _V )
)
128cnvex 6754 . . . . . 6  |-  `' a  e.  _V
13 imaexg 6744 . . . . . 6  |-  ( `' a  e.  _V  ->  ( `' a " c
)  e.  _V )
1412, 13ax-mp 5 . . . . 5  |-  ( `' a " c )  e.  _V
15142a1i 12 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  ( c  e.  (SubGrp `  S )  ->  ( `' a "
c )  e.  _V ) )
16 gimghm 16879 . . . . . . . . 9  |-  ( a  e.  ( R GrpIso  S
)  ->  a  e.  ( R  GrpHom  S ) )
17 ghmima 16854 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
1816, 17sylan 473 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( a " b )  e.  (SubGrp `  S )
)
19 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
2119, 20gimf1o 16878 . . . . . . . . . . 11  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
22 f1of1 5830 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -1-1-> ( Base `  S ) )
2321, 22syl 17 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-> ( Base `  S
) )
2419subgss 16769 . . . . . . . . . 10  |-  ( b  e.  (SubGrp `  R
)  ->  b  C_  ( Base `  R )
)
25 f1imacnv 5847 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -1-1-> ( Base `  S )  /\  b  C_  ( Base `  R
) )  ->  ( `' a " (
a " b ) )  =  b )
2623, 24, 25syl2an 479 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( `' a " ( a "
b ) )  =  b )
2726eqcomd 2437 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  b  =  ( `' a " (
a " b ) ) )
2818, 27jca 534 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( (
a " b )  e.  (SubGrp `  S
)  /\  b  =  ( `' a " (
a " b ) ) ) )
29 eleq1 2501 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  <->  ( a " b )  e.  (SubGrp `  S )
) )
30 imaeq2 5184 . . . . . . . . 9  |-  ( c  =  ( a "
b )  ->  ( `' a " c
)  =  ( `' a " ( a
" b ) ) )
3130eqeq2d 2443 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
b  =  ( `' a " c )  <-> 
b  =  ( `' a " ( a
" b ) ) ) )
3229, 31anbi12d 715 . . . . . . 7  |-  ( c  =  ( a "
b )  ->  (
( c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) )  <-> 
( ( a "
b )  e.  (SubGrp `  S )  /\  b  =  ( `' a
" ( a "
b ) ) ) ) )
3328, 32syl5ibrcom 225 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  b  e.  (SubGrp `  R )
)  ->  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) ) )
3433impr 623 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) )  ->  ( c  e.  (SubGrp `  S )  /\  b  =  ( `' a " c
) ) )
35 ghmpreima 16855 . . . . . . . . 9  |-  ( ( a  e.  ( R 
GrpHom  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
3616, 35sylan 473 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( `' a " c )  e.  (SubGrp `  R )
)
37 f1ofo 5838 . . . . . . . . . . 11  |-  ( a : ( Base `  R
)
-1-1-onto-> ( Base `  S )  ->  a : ( Base `  R ) -onto-> ( Base `  S ) )
3821, 37syl 17 . . . . . . . . . 10  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -onto->
( Base `  S )
)
3920subgss 16769 . . . . . . . . . 10  |-  ( c  e.  (SubGrp `  S
)  ->  c  C_  ( Base `  S )
)
40 foimacnv 5848 . . . . . . . . . 10  |-  ( ( a : ( Base `  R ) -onto-> ( Base `  S )  /\  c  C_  ( Base `  S
) )  ->  (
a " ( `' a " c ) )  =  c )
4138, 39, 40syl2an 479 . . . . . . . . 9  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( a " ( `' a
" c ) )  =  c )
4241eqcomd 2437 . . . . . . . 8  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  c  =  ( a " ( `' a " c
) ) )
4336, 42jca 534 . . . . . . 7  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( ( `' a " c
)  e.  (SubGrp `  R )  /\  c  =  ( a "
( `' a "
c ) ) ) )
44 eleq1 2501 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  <->  ( `' a " c )  e.  (SubGrp `  R )
) )
45 imaeq2 5184 . . . . . . . . 9  |-  ( b  =  ( `' a
" c )  -> 
( a " b
)  =  ( a
" ( `' a
" c ) ) )
4645eqeq2d 2443 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( c  =  ( a " b )  <-> 
c  =  ( a
" ( `' a
" c ) ) ) )
4744, 46anbi12d 715 . . . . . . 7  |-  ( b  =  ( `' a
" c )  -> 
( ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) )  <->  ( ( `' a " c )  e.  (SubGrp `  R
)  /\  c  =  ( a " ( `' a " c
) ) ) ) )
4843, 47syl5ibrcom 225 . . . . . 6  |-  ( ( a  e.  ( R GrpIso  S )  /\  c  e.  (SubGrp `  S )
)  ->  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  /\  c  =  ( a "
b ) ) ) )
4948impr 623 . . . . 5  |-  ( ( a  e.  ( R GrpIso  S )  /\  (
c  e.  (SubGrp `  S )  /\  b  =  ( `' a
" c ) ) )  ->  ( b  e.  (SubGrp `  R )  /\  c  =  (
a " b ) ) )
5034, 49impbida 840 . . . 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 7612 . . 3  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  ~~  (SubGrp `  S
) )
5251exlimiv 1769 . 2  |-  ( E. a  a  e.  ( R GrpIso  S )  -> 
(SubGrp `  R )  ~~  (SubGrp `  S )
)
533, 52sylbi 198 1  |-  ( R 
~=g𝑔  S  ->  (SubGrp `  R )  ~~  (SubGrp `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   _Vcvv 3087    C_ wss 3442   (/)c0 3767   class class class wbr 4426   `'ccnv 4853   "cima 4857   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    ~~ cen 7574   Basecbs 15084  SubGrpcsubg 16762    GrpHom cghm 16831   GrpIso cgim 16872    ~=g𝑔 cgic 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-subg 16765  df-ghm 16832  df-gim 16874  df-gic 16875
This theorem is referenced by: (None)
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