Theorem prdsinvlem 13696

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 5782	. . . . . 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 5483	. . . . . . . 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 13370	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
18	2, 3, 5, 17	grpinvcld 13637	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
19	18	ralrimiva 2605	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	6, 7, 8, 10, 12	prdsbasmpt 13368	. . . 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 5782	. . . . . 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 13370	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
34	23, 24, 25, 26, 27, 33	grplinvd 13643	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
35		2fveq3 5644	. . . . . . . 8 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
36		fveq2 5639	. . . . . . . 8 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	35, 36	fveq12d 5646	. . . . . . 7 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
38	23, 26, 27, 33	grpinvcld 13637	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. ( Base ` ( R ` x ) ) )
39	1, 37, 32, 38	fvmptd3 5740	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
40	39	oveq1d 6033	. . . . 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 5640	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
43		fvco2 5715	. . . . . . 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 4179	. . 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 13371	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
50		fn0g 13463	. . . . . 6 |- 0g Fn _V
51		ssv 3249	. . . . . . 7 |- ran R C_ _V
52	51	a1i 9	. . . . . 6 |- ( ph -> ran R C_ _V )
53		fnco 5440	. . . . . 6 |- ( ( 0g Fn _V /\ R Fn I /\ ran R C_ _V ) -> ( 0g o. R ) Fn I )
54	50, 12, 52, 53	mp3an2i 1378	. . . . 5 |- ( ph -> ( 0g o. R ) Fn I )
55	41	fneq1i 5424	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
56	54, 55	sylibr 134	. . . 4 |- ( ph -> .0. Fn I )
57		dffn5im 5691	. . . 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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   |-> cmpt 4150   ran crn 4726   o. ccom 4729   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6018   Basecbs 13087   +g cplusg 13165   0gc0g 13344   X__scprds 13353   Grpcgrp 13588   invgcminusg 13589

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-ixp 6868  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-struct 13089  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-mulr 13179  df-sca 13181  df-vsca 13182  df-ip 13183  df-tset 13184  df-ple 13185  df-ds 13187  df-hom 13189  df-cco 13190  df-rest 13329  df-topn 13330  df-0g 13346  df-topgen 13348  df-pt 13349  df-prds 13355  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592

This theorem is referenced by:  prdsgrpd  13697  prdsinvgd  13698