Theorem prdsinvlem 13752

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 2231	. . . . . 6 |- ( Base ` ( R ` y ) ) = ( Base ` ( R ` y ) )
3		eqid 2231	. . . . . 6 |- ( invg ` ( R ` y ) ) = ( invg ` ( R ` y ) )
4		prdsinvlem.r	. . . . . . 7 |- ( ph -> R : I --> Grp )
5	4	ffvelcdmda 5790	. . . . . 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 5490	. . . . . . . 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 13426	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
18	2, 3, 5, 17	grpinvcld 13693	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
19	18	ralrimiva 2606	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	6, 7, 8, 10, 12	prdsbasmpt 13424	. . . 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 2318	. 2 |- ( ph -> N e. B )
23		eqid 2231	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
24		eqid 2231	. . . . . 6 |- ( +g ` ( R ` x ) ) = ( +g ` ( R ` x ) )
25		eqid 2231	. . . . . 6 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
26		eqid 2231	. . . . . 6 |- ( invg ` ( R ` x ) ) = ( invg ` ( R ` x ) )
27	4	ffvelcdmda 5790	. . . . . 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 13426	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
34	23, 24, 25, 26, 27, 33	grplinvd 13699	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
35		2fveq3 5653	. . . . . . . 8 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
36		fveq2 5648	. . . . . . . 8 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	35, 36	fveq12d 5655	. . . . . . 7 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
38	23, 26, 27, 33	grpinvcld 13693	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. ( Base ` ( R ` x ) ) )
39	1, 37, 32, 38	fvmptd3 5749	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
40	39	oveq1d 6043	. . . . 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 5649	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
43		fvco2 5724	. . . . . . 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 2276	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .0. ` x ) = ( 0g ` ( R ` x ) ) )
46	34, 40, 45	3eqtr4d 2274	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( .0. ` x ) )
47	46	mpteq2dva 4184	. . 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 13427	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
50		fn0g 13519	. . . . . 6 |- 0g Fn _V
51		ssv 3250	. . . . . . 7 |- ran R C_ _V
52	51	a1i 9	. . . . . 6 |- ( ph -> ran R C_ _V )
53		fnco 5447	. . . . . 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 5431	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
56	54, 55	sylibr 134	. . . 4 |- ( ph -> .0. Fn I )
57		dffn5im 5700	. . . 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 2274	. 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 2202   A.wral 2511   _Vcvv 2803   C_ wss 3201   |-> cmpt 4155   ran crn 4732   o. ccom 4735   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13143   +g cplusg 13221   0gc0g 13400   X__scprds 13409   Grpcgrp 13644   invgcminusg 13645

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-ixp 6911  df-sup 7226  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-fz 10287  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-mulr 13235  df-sca 13237  df-vsca 13238  df-ip 13239  df-tset 13240  df-ple 13241  df-ds 13243  df-hom 13245  df-cco 13246  df-rest 13385  df-topn 13386  df-0g 13402  df-topgen 13404  df-pt 13405  df-prds 13411  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648

This theorem is referenced by:  prdsgrpd  13753  prdsinvgd  13754