Theorem lsmhash 18118

Description: The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
lsmhash.p	|- .(+) = ( LSSum ` G )
lsmhash.o	|- .0. = ( 0g ` G )
lsmhash.z	|- Z = (Cntz ` G )
lsmhash.t	|- ( ph -> T e. (SubGrp ` G ) )
lsmhash.u	|- ( ph -> U e. (SubGrp ` G ) )
lsmhash.i	|- ( ph -> ( T i^i U ) = { .0. } )
lsmhash.s	|- ( ph -> T C_ ( Z ` U ) )
lsmhash.1	|- ( ph -> T e. Fin )
lsmhash.2	|- ( ph -> U e. Fin )

Assertion

Ref	Expression
lsmhash	|- ( ph -> ( # ` ( T .(+) U ) ) = ( ( # ` T ) x. ( # ` U ) ) )

Proof of Theorem lsmhash

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 6680	. . . 4 |- ( ph -> ( T .(+) U ) e. _V )
2		lsmhash.t	. . . . 5 |- ( ph -> T e. (SubGrp ` G ) )
3		lsmhash.u	. . . . 5 |- ( ph -> U e. (SubGrp ` G ) )
4		xpexg 6960	. . . . 5 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( T X. U ) e. _V )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ph -> ( T X. U ) e. _V )
6		eqid 2622	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
7		lsmhash.p	. . . . . . . 8 |- .(+) = ( LSSum ` G )
8		lsmhash.o	. . . . . . . 8 |- .0. = ( 0g ` G )
9		lsmhash.z	. . . . . . . 8 |- Z = (Cntz ` G )
10		lsmhash.i	. . . . . . . 8 |- ( ph -> ( T i^i U ) = { .0. } )
11		lsmhash.s	. . . . . . . 8 |- ( ph -> T C_ ( Z ` U ) )
12		eqid 2622	. . . . . . . 8 |- ( proj1 ` G ) = ( proj1 ` G )
13	6, 7, 8, 9, 2, 3, 10, 11, 12	pj1f 18110	. . . . . . 7 |- ( ph -> ( T ( proj1 ` G ) U ) : ( T .(+) U ) --> T )
14	13	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( T .(+) U ) ) -> ( ( T ( proj1 ` G ) U ) ` x ) e. T )
15	6, 7, 8, 9, 2, 3, 10, 11, 12	pj2f 18111	. . . . . . 7 |- ( ph -> ( U ( proj1 ` G ) T ) : ( T .(+) U ) --> U )
16	15	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( T .(+) U ) ) -> ( ( U ( proj1 ` G ) T ) ` x ) e. U )
17		opelxpi 5148	. . . . . 6 |- ( ( ( ( T ( proj1 ` G ) U ) ` x ) e. T /\ ( ( U ( proj1 ` G ) T ) ` x ) e. U ) -> <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. e. ( T X. U ) )
18	14, 16, 17	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. ( T .(+) U ) ) -> <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. e. ( T X. U ) )
19	18	ex 450	. . . 4 |- ( ph -> ( x e. ( T .(+) U ) -> <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. e. ( T X. U ) ) )
20	2, 3	jca 554	. . . . . 6 |- ( ph -> ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) )
21		xp1st 7198	. . . . . . 7 |- ( y e. ( T X. U ) -> ( 1st ` y ) e. T )
22		xp2nd 7199	. . . . . . 7 |- ( y e. ( T X. U ) -> ( 2nd ` y ) e. U )
23	21, 22	jca 554	. . . . . 6 |- ( y e. ( T X. U ) -> ( ( 1st ` y ) e. T /\ ( 2nd ` y ) e. U ) )
24	6, 7	lsmelvali 18065	. . . . . 6 |- ( ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) /\ ( ( 1st ` y ) e. T /\ ( 2nd ` y ) e. U ) ) -> ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) e. ( T .(+) U ) )
25	20, 23, 24	syl2an 494	. . . . 5 |- ( ( ph /\ y e. ( T X. U ) ) -> ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) e. ( T .(+) U ) )
26	25	ex 450	. . . 4 |- ( ph -> ( y e. ( T X. U ) -> ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) e. ( T .(+) U ) ) )
27	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> T e. (SubGrp ` G ) )
28	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> U e. (SubGrp ` G ) )
29	10	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( T i^i U ) = { .0. } )
30	11	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> T C_ ( Z ` U ) )
31		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> x e. ( T .(+) U ) )
32	21	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( 1st ` y ) e. T )
33	22	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( 2nd ` y ) e. U )
34	6, 7, 8, 9, 27, 28, 29, 30, 12, 31, 32, 33	pj1eq 18113	. . . . . . 7 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( x = ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) <-> ( ( ( T ( proj1 ` G ) U ) ` x ) = ( 1st ` y ) /\ ( ( U ( proj1 ` G ) T ) ` x ) = ( 2nd ` y ) ) ) )
35		eqcom 2629	. . . . . . . 8 |- ( ( ( T ( proj1 ` G ) U ) ` x ) = ( 1st ` y ) <-> ( 1st ` y ) = ( ( T ( proj1 ` G ) U ) ` x ) )
36		eqcom 2629	. . . . . . . 8 |- ( ( ( U ( proj1 ` G ) T ) ` x ) = ( 2nd ` y ) <-> ( 2nd ` y ) = ( ( U ( proj1 ` G ) T ) ` x ) )
37	35, 36	anbi12i 733	. . . . . . 7 |- ( ( ( ( T ( proj1 ` G ) U ) ` x ) = ( 1st ` y ) /\ ( ( U ( proj1 ` G ) T ) ` x ) = ( 2nd ` y ) ) <-> ( ( 1st ` y ) = ( ( T ( proj1 ` G ) U ) ` x ) /\ ( 2nd ` y ) = ( ( U ( proj1 ` G ) T ) ` x ) ) )
38	34, 37	syl6bb 276	. . . . . 6 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( x = ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) <-> ( ( 1st ` y ) = ( ( T ( proj1 ` G ) U ) ` x ) /\ ( 2nd ` y ) = ( ( U ( proj1 ` G ) T ) ` x ) ) ) )
39		eqop 7208	. . . . . . 7 |- ( y e. ( T X. U ) -> ( y = <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. <-> ( ( 1st ` y ) = ( ( T ( proj1 ` G ) U ) ` x ) /\ ( 2nd ` y ) = ( ( U ( proj1 ` G ) T ) ` x ) ) ) )
40	39	ad2antll 765	. . . . . 6 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( y = <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. <-> ( ( 1st ` y ) = ( ( T ( proj1 ` G ) U ) ` x ) /\ ( 2nd ` y ) = ( ( U ( proj1 ` G ) T ) ` x ) ) ) )
41	38, 40	bitr4d 271	. . . . 5 |- ( ( ph /\ ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) ) -> ( x = ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) <-> y = <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. ) )
42	41	ex 450	. . . 4 |- ( ph -> ( ( x e. ( T .(+) U ) /\ y e. ( T X. U ) ) -> ( x = ( ( 1st ` y ) ( +g ` G ) ( 2nd ` y ) ) <-> y = <. ( ( T ( proj1 ` G ) U ) ` x ) , ( ( U ( proj1 ` G ) T ) ` x ) >. ) ) )
43	1, 5, 19, 26, 42	en3d 7992	. . 3 |- ( ph -> ( T .(+) U ) ~~ ( T X. U ) )
44		hasheni 13136	. . 3 |- ( ( T .(+) U ) ~~ ( T X. U ) -> ( # ` ( T .(+) U ) ) = ( # ` ( T X. U ) ) )
45	43, 44	syl 17	. 2 |- ( ph -> ( # ` ( T .(+) U ) ) = ( # ` ( T X. U ) ) )
46		lsmhash.1	. . 3 |- ( ph -> T e. Fin )
47		lsmhash.2	. . 3 |- ( ph -> U e. Fin )
48		hashxp 13221	. . 3 |- ( ( T e. Fin /\ U e. Fin ) -> ( # ` ( T X. U ) ) = ( ( # ` T ) x. ( # ` U ) ) )
49	46, 47, 48	syl2anc 693	. 2 |- ( ph -> ( # ` ( T X. U ) ) = ( ( # ` T ) x. ( # ` U ) ) )
50	45, 49	eqtrd 2656	1 |- ( ph -> ( # ` ( T .(+) U ) ) = ( ( # ` T ) x. ( # ` U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ~~ cen 7952   Fincfn 7955   x. cmul 9941   #chash 13117   +g cplusg 15941   0gc0g 16100  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   proj1cpj1 18050

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-int 4476  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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  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-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-pj1 18052

This theorem is referenced by:  ablfacrp2  18466  ablfac1eulem  18471  ablfac1eu  18472  pgpfaclem2  18481