Theorem prdsinvlem 17524

Description: Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
prdsinvlem.y	|- Y = ( S X_s R )
prdsinvlem.b	|- B = ( Base ` Y )
prdsinvlem.p	|- .+ = ( +g ` Y )
prdsinvlem.s	|- ( ph -> S e. V )
prdsinvlem.i	|- ( ph -> I e. W )
prdsinvlem.r	|- ( ph -> R : I --> Grp )
prdsinvlem.f	|- ( ph -> F e. B )
prdsinvlem.z	|- .0. = ( 0g o. R )
prdsinvlem.n	|- N = ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) )

Assertion

Ref	Expression
prdsinvlem	|- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )

Distinct variable groups:   y, B   y, F   y, I   ph, y   y, R   y, S   y, V   y, W   y, Y

Allowed substitution hints:   .+ ( y)   N( y)   .0. ( y)

Proof of Theorem prdsinvlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsinvlem.n	. . 3 |- N = ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) )
2		prdsinvlem.r	. . . . . . 7 |- ( ph -> R : I --> Grp )
3	2	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( R ` y ) e. Grp )
4		prdsinvlem.y	. . . . . . 7 |- Y = ( S X_s R )
5		prdsinvlem.b	. . . . . . 7 |- B = ( Base ` Y )
6		prdsinvlem.s	. . . . . . . 8 |- ( ph -> S e. V )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. I ) -> S e. V )
8		prdsinvlem.i	. . . . . . . 8 |- ( ph -> I e. W )
9	8	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. I ) -> I e. W )
10		ffn 6045	. . . . . . . . 9 |- ( R : I --> Grp -> R Fn I )
11	2, 10	syl 17	. . . . . . . 8 |- ( ph -> R Fn I )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. I ) -> R Fn I )
13		prdsinvlem.f	. . . . . . . 8 |- ( ph -> F e. B )
14	13	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. I ) -> F e. B )
15		simpr 477	. . . . . . 7 |- ( ( ph /\ y e. I ) -> y e. I )
16	4, 5, 7, 9, 12, 14, 15	prdsbasprj 16132	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
17		eqid 2622	. . . . . . 7 |- ( Base ` ( R ` y ) ) = ( Base ` ( R ` y ) )
18		eqid 2622	. . . . . . 7 |- ( invg ` ( R ` y ) ) = ( invg ` ( R ` y ) )
19	17, 18	grpinvcl 17467	. . . . . 6 |- ( ( ( R ` y ) e. Grp /\ ( F ` y ) e. ( Base ` ( R ` y ) ) ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	3, 16, 19	syl2anc 693	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
21	20	ralrimiva 2966	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
22	4, 5, 6, 8, 11	prdsbasmpt 16130	. . . 4 |- ( ph -> ( ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) ) e. B <-> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) ) )
23	21, 22	mpbird 247	. . 3 |- ( ph -> ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) ) e. B )
24	1, 23	syl5eqel 2705	. 2 |- ( ph -> N e. B )
25	2	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( R ` x ) e. Grp )
26	6	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> S e. V )
27	8	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> I e. W )
28	11	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> R Fn I )
29	13	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> F e. B )
30		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. I ) -> x e. I )
31	4, 5, 26, 27, 28, 29, 30	prdsbasprj 16132	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
32		eqid 2622	. . . . . . 7 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
33		eqid 2622	. . . . . . 7 |- ( +g ` ( R ` x ) ) = ( +g ` ( R ` x ) )
34		eqid 2622	. . . . . . 7 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
35		eqid 2622	. . . . . . 7 |- ( invg ` ( R ` x ) ) = ( invg ` ( R ` x ) )
36	32, 33, 34, 35	grplinv 17468	. . . . . 6 |- ( ( ( R ` x ) e. Grp /\ ( F ` x ) e. ( Base ` ( R ` x ) ) ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
37	25, 31, 36	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
38		fveq2 6191	. . . . . . . . . 10 |- ( y = x -> ( R ` y ) = ( R ` x ) )
39	38	fveq2d 6195	. . . . . . . . 9 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
40		fveq2 6191	. . . . . . . . 9 |- ( y = x -> ( F ` y ) = ( F ` x ) )
41	39, 40	fveq12d 6197	. . . . . . . 8 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
42		fvex 6201	. . . . . . . 8 |- ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. _V
43	41, 1, 42	fvmpt 6282	. . . . . . 7 |- ( x e. I -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
44	43	adantl 482	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
45	44	oveq1d 6665	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) )
46		prdsinvlem.z	. . . . . . 7 |- .0. = ( 0g o. R )
47	46	fveq1i 6192	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
48		fvco2 6273	. . . . . . 7 |- ( ( R Fn I /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
49	11, 48	sylan 488	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
50	47, 49	syl5eq 2668	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .0. ` x ) = ( 0g ` ( R ` x ) ) )
51	37, 45, 50	3eqtr4d 2666	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( .0. ` x ) )
52	51	mpteq2dva 4744	. . 3 |- ( ph -> ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) = ( x e. I |-> ( .0. ` x ) ) )
53		prdsinvlem.p	. . . 4 |- .+ = ( +g ` Y )
54	4, 5, 6, 8, 11, 24, 13, 53	prdsplusgval 16133	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
55		fn0g 17262	. . . . . . 7 |- 0g Fn _V
56	55	a1i 11	. . . . . 6 |- ( ph -> 0g Fn _V )
57		ssv 3625	. . . . . . 7 |- ran R C_ _V
58	57	a1i 11	. . . . . 6 |- ( ph -> ran R C_ _V )
59		fnco 5999	. . . . . 6 |- ( ( 0g Fn _V /\ R Fn I /\ ran R C_ _V ) -> ( 0g o. R ) Fn I )
60	56, 11, 58, 59	syl3anc 1326	. . . . 5 |- ( ph -> ( 0g o. R ) Fn I )
61	46	fneq1i 5985	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
62	60, 61	sylibr 224	. . . 4 |- ( ph -> .0. Fn I )
63		dffn5 6241	. . . 4 |- ( .0. Fn I <-> .0. = ( x e. I |-> ( .0. ` x ) ) )
64	62, 63	sylib 208	. . 3 |- ( ph -> .0. = ( x e. I |-> ( .0. ` x ) ) )
65	52, 54, 64	3eqtr4d 2666	. 2 |- ( ph -> ( N .+ F ) = .0. )
66	24, 65	jca 554	1 |- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   X__scprds 16106   Grpcgrp 17422   invgcminusg 17423

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-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  prdsgrpd  17525  prdsinvgd  17526