Theorem prdsinvlem 13627

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		eqid 2229	. . . . . 6 |- ( Base ` ( R ` y ) ) = ( Base ` ( R ` y ) )
3		eqid 2229	. . . . . 6 |- ( invg ` ( R ` y ) ) = ( invg ` ( R ` y ) )
4		prdsinvlem.r	. . . . . . 7 |- ( ph -> R : I --> Grp )
5	4	ffvelcdmda 5763	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( R ` y ) e. Grp )
6		prdsinvlem.y	. . . . . . 7 |- Y = ( S X_s R )
7		prdsinvlem.b	. . . . . . 7 |- B = ( Base ` Y )
8		prdsinvlem.s	. . . . . . . 8 |- ( ph -> S e. V )
9	8	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> S e. V )
10		prdsinvlem.i	. . . . . . . 8 |- ( ph -> I e. W )
11	10	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> I e. W )
12	4	ffnd 5470	. . . . . . . 8 |- ( ph -> R Fn I )
13	12	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> R Fn I )
14		prdsinvlem.f	. . . . . . . 8 |- ( ph -> F e. B )
15	14	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> F e. B )
16		simpr 110	. . . . . . 7 |- ( ( ph /\ y e. I ) -> y e. I )
17	6, 7, 9, 11, 13, 15, 16	prdsbasprj 13301	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
18	2, 3, 5, 17	grpinvcld 13568	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
19	18	ralrimiva 2603	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	6, 7, 8, 10, 12	prdsbasmpt 13299	. . . 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 ) ) ) )
21	19, 20	mpbird 167	. . 3 |- ( ph -> ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) ) e. B )
22	1, 21	eqeltrid 2316	. 2 |- ( ph -> N e. B )
23		eqid 2229	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
24		eqid 2229	. . . . . 6 |- ( +g ` ( R ` x ) ) = ( +g ` ( R ` x ) )
25		eqid 2229	. . . . . 6 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
26		eqid 2229	. . . . . 6 |- ( invg ` ( R ` x ) ) = ( invg ` ( R ` x ) )
27	4	ffvelcdmda 5763	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( R ` x ) e. Grp )
28	8	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. I ) -> S e. V )
29	10	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. I ) -> I e. W )
30	12	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. I ) -> R Fn I )
31	14	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. I ) -> F e. B )
32		simpr 110	. . . . . . 7 |- ( ( ph /\ x e. I ) -> x e. I )
33	6, 7, 28, 29, 30, 31, 32	prdsbasprj 13301	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
34	23, 24, 25, 26, 27, 33	grplinvd 13574	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
35		2fveq3 5628	. . . . . . . 8 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
36		fveq2 5623	. . . . . . . 8 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	35, 36	fveq12d 5630	. . . . . . 7 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
38	23, 26, 27, 33	grpinvcld 13568	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. ( Base ` ( R ` x ) ) )
39	1, 37, 32, 38	fvmptd3 5721	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
40	39	oveq1d 6009	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) )
41		prdsinvlem.z	. . . . . . 7 |- .0. = ( 0g o. R )
42	41	fveq1i 5624	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
43		fvco2 5696	. . . . . . 7 |- ( ( R Fn I /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
44	12, 43	sylan 283	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
45	42, 44	eqtrid 2274	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .0. ` x ) = ( 0g ` ( R ` x ) ) )
46	34, 40, 45	3eqtr4d 2272	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( .0. ` x ) )
47	46	mpteq2dva 4173	. . 3 |- ( ph -> ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) = ( x e. I |-> ( .0. ` x ) ) )
48		prdsinvlem.p	. . . 4 |- .+ = ( +g ` Y )
49	6, 7, 8, 10, 12, 22, 14, 48	prdsplusgval 13302	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
50		fn0g 13394	. . . . . 6 |- 0g Fn _V
51		ssv 3246	. . . . . . 7 |- ran R C_ _V
52	51	a1i 9	. . . . . 6 |- ( ph -> ran R C_ _V )
53		fnco 5427	. . . . . 6 |- ( ( 0g Fn _V /\ R Fn I /\ ran R C_ _V ) -> ( 0g o. R ) Fn I )
54	50, 12, 52, 53	mp3an2i 1376	. . . . 5 |- ( ph -> ( 0g o. R ) Fn I )
55	41	fneq1i 5411	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
56	54, 55	sylibr 134	. . . 4 |- ( ph -> .0. Fn I )
57		dffn5im 5672	. . . 4 |- ( .0. Fn I -> .0. = ( x e. I |-> ( .0. ` x ) ) )
58	56, 57	syl 14	. . 3 |- ( ph -> .0. = ( x e. I |-> ( .0. ` x ) ) )
59	47, 49, 58	3eqtr4d 2272	. 2 |- ( ph -> ( N .+ F ) = .0. )
60	22, 59	jca 306	1 |- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   |-> cmpt 4144   ran crn 4717   o. ccom 4720   Fn wfn 5309   -->wf 5310   ` cfv 5314  (class class class)co 5994   Basecbs 13018   +g cplusg 13096   0gc0g 13275   X__scprds 13284   Grpcgrp 13519   invgcminusg 13520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-map 6787  df-ixp 6836  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-fz 10193  df-struct 13020  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-mulr 13110  df-sca 13112  df-vsca 13113  df-ip 13114  df-tset 13115  df-ple 13116  df-ds 13118  df-hom 13120  df-cco 13121  df-rest 13260  df-topn 13261  df-0g 13277  df-topgen 13279  df-pt 13280  df-prds 13286  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-minusg 13523

This theorem is referenced by:  prdsgrpd  13628  prdsinvgd  13629