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Theorem gicsubgen 14742
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 14733 . . 3  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3464 . . 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 5539 . . . . 5  |-  (SubGrp `  R )  e.  _V
54a1i 10 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  e.  _V )
6 fvex 5539 . . . . 5  |-  (SubGrp `  S )  e.  _V
76a1i 10 . . . 4  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  S
)  e.  _V )
8 vex 2791 . . . . . 6  |-  a  e. 
_V
9 imaexg 5026 . . . . . 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 5209 . . . . . 6  |-  `' a  e.  _V
13 imaexg 5026 . . . . . 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 14728 . . . . . . . . 9  |-  ( a  e.  ( R GrpIso  S
)  ->  a  e.  ( R  GrpHom  S ) )
17 ghmima 14703 . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
2119, 20gimf1o 14727 . . . . . . . . . . 11  |-  ( a  e.  ( R GrpIso  S
)  ->  a :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
22 f1of1 5471 . . . . . . . . . . 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 14622 . . . . . . . . . 10  |-  ( b  e.  (SubGrp `  R
)  ->  b  C_  ( Base `  R )
)
25 f1imacnv 5489 . . . . . . . . . 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 2288 . . . . . . . 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 2343 . . . . . . . 8  |-  ( c  =  ( a "
b )  ->  (
c  e.  (SubGrp `  S )  <->  ( a " b )  e.  (SubGrp `  S )
) )
30 imaeq2 5008 . . . . . . . . 9  |-  ( c  =  ( a "
b )  ->  ( `' a " c
)  =  ( `' a " ( a
" b ) ) )
3130eqeq2d 2294 . . . . . . . 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 14704 . . . . . . . . 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 5479 . . . . . . . . . . 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 14622 . . . . . . . . . 10  |-  ( c  e.  (SubGrp `  S
)  ->  c  C_  ( Base `  S )
)
40 foimacnv 5490 . . . . . . . . . 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 2288 . . . . . . . 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 2343 . . . . . . . 8  |-  ( b  =  ( `' a
" c )  -> 
( b  e.  (SubGrp `  R )  <->  ( `' a " c )  e.  (SubGrp `  R )
) )
45 imaeq2 5008 . . . . . . . . 9  |-  ( b  =  ( `' a
" c )  -> 
( a " b
)  =  ( a
" ( `' a
" c ) ) )
4645eqeq2d 2294 . . . . . . . 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 6897 . . 3  |-  ( a  e.  ( R GrpIso  S
)  ->  (SubGrp `  R
)  ~~  (SubGrp `  S
) )
5251exlimiv 1666 . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023   `'ccnv 4688   "cima 4692   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Basecbs 13148  SubGrpcsubg 14615    GrpHom cghm 14680   GrpIso cgim 14721    ~=ph𝑔 cgic 14722
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-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-iun 3907  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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-ghm 14681  df-gim 14723  df-gic 14724
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