Theorem prdsinvlem 13813

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 2232	. . . . . 6 |- ( Base ` ( R ` y ) ) = ( Base ` ( R ` y ) )
3		eqid 2232	. . . . . 6 |- ( invg ` ( R ` y ) ) = ( invg ` ( R ` y ) )
4		prdsinvlem.r	. . . . . . 7 |- ( ph -> R : I --> Grp )
5	4	ffvelcdmda 5811	. . . . . 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 5508	. . . . . . . 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 13487	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. ( Base ` ( R ` y ) ) )
18	2, 3, 5, 17	grpinvcld 13754	. . . . 5 |- ( ( ph /\ y e. I ) -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
19	18	ralrimiva 2615	. . . 4 |- ( ph -> A. y e. I ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) e. ( Base ` ( R ` y ) ) )
20	6, 7, 8, 10, 12	prdsbasmpt 13485	. . . 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 2319	. 2 |- ( ph -> N e. B )
23		eqid 2232	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
24		eqid 2232	. . . . . 6 |- ( +g ` ( R ` x ) ) = ( +g ` ( R ` x ) )
25		eqid 2232	. . . . . 6 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
26		eqid 2232	. . . . . 6 |- ( invg ` ( R ` x ) ) = ( invg ` ( R ` x ) )
27	4	ffvelcdmda 5811	. . . . . 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 13487	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( Base ` ( R ` x ) ) )
34	23, 24, 25, 26, 27, 33	grplinvd 13760	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( 0g ` ( R ` x ) ) )
35		2fveq3 5674	. . . . . . . 8 |- ( y = x -> ( invg ` ( R ` y ) ) = ( invg ` ( R ` x ) ) )
36		fveq2 5669	. . . . . . . 8 |- ( y = x -> ( F ` y ) = ( F ` x ) )
37	35, 36	fveq12d 5676	. . . . . . 7 |- ( y = x -> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
38	23, 26, 27, 33	grpinvcld 13754	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) e. ( Base ` ( R ` x ) ) )
39	1, 37, 32, 38	fvmptd3 5770	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( N ` x ) = ( ( invg ` ( R ` x ) ) ` ( F ` x ) ) )
40	39	oveq1d 6064	. . . . 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 5670	. . . . . 6 |- ( .0. ` x ) = ( ( 0g o. R ) ` x )
43		fvco2 5745	. . . . . . 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 2277	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .0. ` x ) = ( 0g ` ( R ` x ) ) )
46	34, 40, 45	3eqtr4d 2275	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) = ( .0. ` x ) )
47	46	mpteq2dva 4199	. . 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 13488	. . 3 |- ( ph -> ( N .+ F ) = ( x e. I |-> ( ( N ` x ) ( +g ` ( R ` x ) ) ( F ` x ) ) ) )
50		fn0g 13580	. . . . . 6 |- 0g Fn _V
51		ssv 3259	. . . . . . 7 |- ran R C_ _V
52	51	a1i 9	. . . . . 6 |- ( ph -> ran R C_ _V )
53		fnco 5465	. . . . . 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 5449	. . . . 5 |- ( .0. Fn I <-> ( 0g o. R ) Fn I )
56	54, 55	sylibr 134	. . . 4 |- ( ph -> .0. Fn I )
57		dffn5im 5721	. . . 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 2275	. 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 2203   A.wral 2520   _Vcvv 2812   C_ wss 3210   |-> cmpt 4170   ran crn 4749   o. ccom 4752   Fn wfn 5346   -->wf 5347   ` cfv 5351  (class class class)co 6049   Basecbs 13204   +g cplusg 13282   0gc0g 13461   X__scprds 13470   Grpcgrp 13705   invgcminusg 13706

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-ixp 6933  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-z 9577  df-dec 9709  df-uz 9853  df-fz 10342  df-struct 13206  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-tset 13301  df-ple 13302  df-ds 13304  df-hom 13306  df-cco 13307  df-rest 13446  df-topn 13447  df-0g 13463  df-topgen 13465  df-pt 13466  df-prds 13472  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709

This theorem is referenced by:  prdsgrpd  13814  prdsinvgd  13815