Theorem gicsubgen 17721

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.

Step	Hyp	Ref	Expression
1		brgic 17711	. . 3 |- ( R ~=g𝑔 S <-> ( R GrpIso S ) =/= (/) )
2		n0 3931	. . 3 |- ( ( R GrpIso S ) =/= (/) <-> E. a a e. ( R GrpIso S ) )
3	1, 2	bitri 264	. 2 |- ( R ~=g𝑔 S <-> E. a a e. ( R GrpIso S ) )
4		fvexd 6203	. . . 4 |- ( a e. ( R GrpIso S ) -> (SubGrp ` R ) e. _V )
5		fvexd 6203	. . . 4 |- ( a e. ( R GrpIso S ) -> (SubGrp ` S ) e. _V )
6		vex 3203	. . . . . 6 |- a e. _V
7	6	imaex 7104	. . . . 5 |- ( a " b ) e. _V
8	7	2a1i 12	. . . 4 |- ( a e. ( R GrpIso S ) -> ( b e. (SubGrp ` R ) -> ( a " b ) e. _V ) )
9	6	cnvex 7113	. . . . . 6 |- `' a e. _V
10	9	imaex 7104	. . . . 5 |- ( `' a " c ) e. _V
11	10	2a1i 12	. . . 4 |- ( a e. ( R GrpIso S ) -> ( c e. (SubGrp ` S ) -> ( `' a " c ) e. _V ) )
12		gimghm 17706	. . . . . . . . 9 |- ( a e. ( R GrpIso S ) -> a e. ( R GrpHom S ) )
13		ghmima 17681	. . . . . . . . 9 |- ( ( a e. ( R GrpHom S ) /\ b e. (SubGrp ` R ) ) -> ( a " b ) e. (SubGrp ` S ) )
14	12, 13	sylan 488	. . . . . . . 8 |- ( ( a e. ( R GrpIso S ) /\ b e. (SubGrp ` R ) ) -> ( a " b ) e. (SubGrp ` S ) )
15		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
16		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` S ) = ( Base ` S )
17	15, 16	gimf1o 17705	. . . . . . . . . . 11 |- ( a e. ( R GrpIso S ) -> a : ( Base ` R ) -1-1-onto-> ( Base ` S ) )
18		f1of1 6136	. . . . . . . . . . 11 |- ( a : ( Base ` R ) -1-1-onto-> ( Base ` S ) -> a : ( Base ` R ) -1-1-> ( Base ` S ) )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( a e. ( R GrpIso S ) -> a : ( Base ` R ) -1-1-> ( Base ` S ) )
20	15	subgss 17595	. . . . . . . . . 10 |- ( b e. (SubGrp ` R ) -> b C_ ( Base ` R ) )
21		f1imacnv 6153	. . . . . . . . . 10 |- ( ( a : ( Base ` R ) -1-1-> ( Base ` S ) /\ b C_ ( Base ` R ) ) -> ( `' a " ( a " b ) ) = b )
22	19, 20, 21	syl2an 494	. . . . . . . . 9 |- ( ( a e. ( R GrpIso S ) /\ b e. (SubGrp ` R ) ) -> ( `' a " ( a " b ) ) = b )
23	22	eqcomd 2628	. . . . . . . 8 |- ( ( a e. ( R GrpIso S ) /\ b e. (SubGrp ` R ) ) -> b = ( `' a " ( a " b ) ) )
24	14, 23	jca 554	. . . . . . 7 |- ( ( a e. ( R GrpIso S ) /\ b e. (SubGrp ` R ) ) -> ( ( a " b ) e. (SubGrp ` S ) /\ b = ( `' a " ( a " b ) ) ) )
25		eleq1 2689	. . . . . . . 8 |- ( c = ( a " b ) -> ( c e. (SubGrp ` S ) <-> ( a " b ) e. (SubGrp ` S ) ) )
26		imaeq2 5462	. . . . . . . . 9 |- ( c = ( a " b ) -> ( `' a " c ) = ( `' a " ( a " b ) ) )
27	26	eqeq2d 2632	. . . . . . . 8 |- ( c = ( a " b ) -> ( b = ( `' a " c ) <-> b = ( `' a " ( a " b ) ) ) )
28	25, 27	anbi12d 747	. . . . . . 7 |- ( c = ( a " b ) -> ( ( c e. (SubGrp ` S ) /\ b = ( `' a " c ) ) <-> ( ( a " b ) e. (SubGrp ` S ) /\ b = ( `' a " ( a " b ) ) ) ) )
29	24, 28	syl5ibrcom 237	. . . . . 6 |- ( ( a e. ( R GrpIso S ) /\ b e. (SubGrp ` R ) ) -> ( c = ( a " b ) -> ( c e. (SubGrp ` S ) /\ b = ( `' a " c ) ) ) )
30	29	impr 649	. . . . 5 |- ( ( a e. ( R GrpIso S ) /\ ( b e. (SubGrp ` R ) /\ c = ( a " b ) ) ) -> ( c e. (SubGrp ` S ) /\ b = ( `' a " c ) ) )
31		ghmpreima 17682	. . . . . . . . 9 |- ( ( a e. ( R GrpHom S ) /\ c e. (SubGrp ` S ) ) -> ( `' a " c ) e. (SubGrp ` R ) )
32	12, 31	sylan 488	. . . . . . . 8 |- ( ( a e. ( R GrpIso S ) /\ c e. (SubGrp ` S ) ) -> ( `' a " c ) e. (SubGrp ` R ) )
33		f1ofo 6144	. . . . . . . . . . 11 |- ( a : ( Base ` R ) -1-1-onto-> ( Base ` S ) -> a : ( Base ` R ) -onto-> ( Base ` S ) )
34	17, 33	syl 17	. . . . . . . . . 10 |- ( a e. ( R GrpIso S ) -> a : ( Base ` R ) -onto-> ( Base ` S ) )
35	16	subgss 17595	. . . . . . . . . 10 |- ( c e. (SubGrp ` S ) -> c C_ ( Base ` S ) )
36		foimacnv 6154	. . . . . . . . . 10 |- ( ( a : ( Base ` R ) -onto-> ( Base ` S ) /\ c C_ ( Base ` S ) ) -> ( a " ( `' a " c ) ) = c )
37	34, 35, 36	syl2an 494	. . . . . . . . 9 |- ( ( a e. ( R GrpIso S ) /\ c e. (SubGrp ` S ) ) -> ( a " ( `' a " c ) ) = c )
38	37	eqcomd 2628	. . . . . . . 8 |- ( ( a e. ( R GrpIso S ) /\ c e. (SubGrp ` S ) ) -> c = ( a " ( `' a " c ) ) )
39	32, 38	jca 554	. . . . . . 7 |- ( ( a e. ( R GrpIso S ) /\ c e. (SubGrp ` S ) ) -> ( ( `' a " c ) e. (SubGrp ` R ) /\ c = ( a " ( `' a " c ) ) ) )
40		eleq1 2689	. . . . . . . 8 |- ( b = ( `' a " c ) -> ( b e. (SubGrp ` R ) <-> ( `' a " c ) e. (SubGrp ` R ) ) )
41		imaeq2 5462	. . . . . . . . 9 |- ( b = ( `' a " c ) -> ( a " b ) = ( a " ( `' a " c ) ) )
42	41	eqeq2d 2632	. . . . . . . 8 |- ( b = ( `' a " c ) -> ( c = ( a " b ) <-> c = ( a " ( `' a " c ) ) ) )
43	40, 42	anbi12d 747	. . . . . . 7 |- ( b = ( `' a " c ) -> ( ( b e. (SubGrp ` R ) /\ c = ( a " b ) ) <-> ( ( `' a " c ) e. (SubGrp ` R ) /\ c = ( a " ( `' a " c ) ) ) ) )
44	39, 43	syl5ibrcom 237	. . . . . 6 |- ( ( a e. ( R GrpIso S ) /\ c e. (SubGrp ` S ) ) -> ( b = ( `' a " c ) -> ( b e. (SubGrp ` R ) /\ c = ( a " b ) ) ) )
45	44	impr 649	. . . . 5 |- ( ( a e. ( R GrpIso S ) /\ ( c e. (SubGrp ` S ) /\ b = ( `' a " c ) ) ) -> ( b e. (SubGrp ` R ) /\ c = ( a " b ) ) )
46	30, 45	impbida 877	. . . 4 |- ( a e. ( R GrpIso S ) -> ( ( b e. (SubGrp ` R ) /\ c = ( a " b ) ) <-> ( c e. (SubGrp ` S ) /\ b = ( `' a " c ) ) ) )
47	4, 5, 8, 11, 46	en2d 7991	. . 3 |- ( a e. ( R GrpIso S ) -> (SubGrp ` R ) ~~ (SubGrp ` S ) )
48	47	exlimiv 1858	. 2 |- ( E. a a e. ( R GrpIso S ) -> (SubGrp ` R ) ~~ (SubGrp ` S ) )
49	3, 48	sylbi 207	1 |- ( R ~=g𝑔 S -> (SubGrp ` R ) ~~ (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   `'ccnv 5113   "cima 5117   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Basecbs 15857  SubGrpcsubg 17588   GrpHom cghm 17657   GrpIso cgim 17699   ~=g_𝑔 cgic 17700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-gim 17701  df-gic 17702

This theorem is referenced by: (None)