Theorem dprddisj 18408

Description: The function S is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
dprdcntz.1	|- ( ph -> G dom DProd S )
dprdcntz.2	|- ( ph -> dom S = I )
dprdcntz.3	|- ( ph -> X e. I )
dprddisj.0	|- .0. = ( 0g ` G )
dprddisj.k	|- K = (mrCls ` (SubGrp ` G ) )

Assertion

Ref	Expression
dprddisj	|- ( ph -> ( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) = { .0. } )

Proof of Theorem dprddisj

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprdcntz.3	. 2 |- ( ph -> X e. I )
2		dprdcntz.1	. . . . 5 |- ( ph -> G dom DProd S )
3		dprdcntz.2	. . . . . . 7 |- ( ph -> dom S = I )
4	2, 3	dprddomcld 18400	. . . . . 6 |- ( ph -> I e. _V )
5		eqid 2622	. . . . . . 7 |- (Cntz ` G ) = (Cntz ` G )
6		dprddisj.0	. . . . . . 7 |- .0. = ( 0g ` G )
7		dprddisj.k	. . . . . . 7 |- K = (mrCls ` (SubGrp ` G ) )
8	5, 6, 7	dmdprd 18397	. . . . . 6 |- ( ( I e. _V /\ dom S = I ) -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
9	4, 3, 8	syl2anc 693	. . . . 5 |- ( ph -> ( G dom DProd S <-> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) ) )
10	2, 9	mpbid 222	. . . 4 |- ( ph -> ( G e. Grp /\ S : I --> (SubGrp ` G ) /\ A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) ) )
11	10	simp3d 1075	. . 3 |- ( ph -> A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) )
12		simpr 477	. . . 4 |- ( ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) -> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } )
13	12	ralimi 2952	. . 3 |- ( A. x e. I ( A. y e. ( I \ { x } ) ( S ` x ) C_ ( (Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } ) -> A. x e. I ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } )
14	11, 13	syl 17	. 2 |- ( ph -> A. x e. I ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } )
15		fveq2 6191	. . . . 5 |- ( x = X -> ( S ` x ) = ( S ` X ) )
16		sneq 4187	. . . . . . . . 9 |- ( x = X -> { x } = { X } )
17	16	difeq2d 3728	. . . . . . . 8 |- ( x = X -> ( I \ { x } ) = ( I \ { X } ) )
18	17	imaeq2d 5466	. . . . . . 7 |- ( x = X -> ( S " ( I \ { x } ) ) = ( S " ( I \ { X } ) ) )
19	18	unieqd 4446	. . . . . 6 |- ( x = X -> U. ( S " ( I \ { x } ) ) = U. ( S " ( I \ { X } ) ) )
20	19	fveq2d 6195	. . . . 5 |- ( x = X -> ( K ` U. ( S " ( I \ { x } ) ) ) = ( K ` U. ( S " ( I \ { X } ) ) ) )
21	15, 20	ineq12d 3815	. . . 4 |- ( x = X -> ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = ( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) )
22	21	eqeq1d 2624	. . 3 |- ( x = X -> ( ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } <-> ( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) = { .0. } ) )
23	22	rspcv 3305	. 2 |- ( X e. I -> ( A. x e. I ( ( S ` x ) i^i ( K ` U. ( S " ( I \ { x } ) ) ) ) = { .0. } -> ( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) = { .0. } ) )
24	1, 14, 23	sylc 65	1 |- ( ph -> ( ( S ` X ) i^i ( K ` U. ( S " ( I \ { X } ) ) ) ) = { .0. } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   class class class wbr 4653   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888   0gc0g 16100  mrClscmrc 16243   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ixp 7909  df-dprd 18394

This theorem is referenced by:  dprdfeq0  18421  dprdres  18427  dprdss  18428  dprdf1o  18431  dprd2da  18441  dmdprdsplit2lem  18444