Theorem prdsinvlem 13905

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 2234	. . . . . 6 |- ( Base ` ( R ` y ) ) = ( Base ` ( R ` y ) )
3		eqid 2234	. . . . . 6 |- ( invg ` ( R ` y ) ) = ( invg ` ( R ` y ) )
4		prdsinvlem.r	. . . . . . 7 |- ( ph -> R : I --> Grp )
5	4	ffvelcdmda 5817	. . . . . 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 5514	. . . . . . . 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 13579	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
18	2, 3, 5, 17	grpinvcld 13846	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
19	18	ralrimiva 2617	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	6, 7, 8, 10, 12	prdsbasmpt 13577	. . . 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 2321	. 2 |- ( ph -> N e. B )
23		eqid 2234	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
24		eqid 2234	. . . . . 6 |- ( +g ` ( R ` x ) ) = ( +g ` ( R ` x ) )
25		eqid 2234	. . . . . 6 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
26		eqid 2234	. . . . . 6 |- ( invg ` ( R ` x ) ) = ( invg ` ( R ` x ) )
27	4	ffvelcdmda 5817	. . . . . 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 13579	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
34	23, 24, 25, 26, 27, 33	grplinvd 13852	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
35		2fveq3 5680	. . . . . . . 8 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
36		fveq2 5675	. . . . . . . 8 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	35, 36	fveq12d 5682	. . . . . . 7 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
38	23, 26, 27, 33	grpinvcld 13846	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. ( Base ` ( R ` x ) ) )
39	1, 37, 32, 38	fvmptd3 5776	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
40	39	oveq1d 6073	. . . . 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 5676	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
43		fvco2 5751	. . . . . . 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 2279	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .0. ` x ) = ( 0g ` ( R ` x ) ) )
46	34, 40, 45	3eqtr4d 2277	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( .0. ` x ) )
47	46	mpteq2dva 4205	. . 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 13580	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
50		fn0g 13672	. . . . . 6 |- 0g Fn _V
51		ssv 3264	. . . . . . 7 |- ran R C_ _V
52	51	a1i 9	. . . . . 6 |- ( ph -> ran R C_ _V )
53		fnco 5471	. . . . . 6 |- ( ( 0g Fn _V /\ R Fn I /\ ran R C_ _V ) -> ( 0g o. R ) Fn I )
54	50, 12, 52, 53	mp3an2i 1379	. . . . 5 |- ( ph -> ( 0g o. R ) Fn I )
55	41	fneq1i 5455	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
56	54, 55	sylibr 134	. . . 4 |- ( ph -> .0. Fn I )
57		dffn5im 5727	. . . 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 2277	. 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 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   |-> cmpt 4176   ran crn 4755   o. ccom 4758   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   X__scprds 13562   Grpcgrp 13797   invgcminusg 13798

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-ixp 6947  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-ip 13392  df-tset 13393  df-ple 13394  df-ds 13396  df-hom 13398  df-cco 13399  df-rest 13538  df-topn 13539  df-0g 13555  df-topgen 13557  df-pt 13558  df-prds 13564  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801

This theorem is referenced by:  prdsgrpd  13906  prdsinvgd  13907