Theorem rrgsupp 19291

Description: Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)

Hypotheses

Ref	Expression
rrgval.e	|- E = (RLReg ` R )
rrgval.b	|- B = ( Base ` R )
rrgval.t	|- .x. = ( .r ` R )
rrgval.z	|- .0. = ( 0g ` R )
rrgsupp.i	|- ( ph -> I e. V )
rrgsupp.r	|- ( ph -> R e. Ring )
rrgsupp.x	|- ( ph -> X e. E )
rrgsupp.y	|- ( ph -> Y : I --> B )

Assertion

Ref	Expression
rrgsupp	|- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = ( Y supp .0. ) )

Proof of Theorem rrgsupp

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

Step	Hyp	Ref	Expression
1		rrgsupp.i	. . . . . . . . 9 |- ( ph -> I e. V )
2		rrgsupp.x	. . . . . . . . . 10 |- ( ph -> X e. E )
3	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. I ) -> X e. E )
4		fvexd 6203	. . . . . . . . 9 |- ( ( ph /\ y e. I ) -> ( Y ` y ) e. _V )
5		fconstmpt 5163	. . . . . . . . . 10 |- ( I X. { X } ) = ( y e. I |-> X )
6	5	a1i 11	. . . . . . . . 9 |- ( ph -> ( I X. { X } ) = ( y e. I |-> X ) )
7		rrgsupp.y	. . . . . . . . . 10 |- ( ph -> Y : I --> B )
8	7	feqmptd 6249	. . . . . . . . 9 |- ( ph -> Y = ( y e. I |-> ( Y ` y ) ) )
9	1, 3, 4, 6, 8	offval2 6914	. . . . . . . 8 |- ( ph -> ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) )
11	10	fveq1d 6193	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) )
12		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. I ) -> x e. I )
13		ovex 6678	. . . . . . 7 |- ( X .x. ( Y ` x ) ) e. _V
14		fveq2 6191	. . . . . . . . 9 |- ( y = x -> ( Y ` y ) = ( Y ` x ) )
15	14	oveq2d 6666	. . . . . . . 8 |- ( y = x -> ( X .x. ( Y ` y ) ) = ( X .x. ( Y ` x ) ) )
16		eqid 2622	. . . . . . . 8 |- ( y e. I |-> ( X .x. ( Y ` y ) ) ) = ( y e. I |-> ( X .x. ( Y ` y ) ) )
17	15, 16	fvmptg 6280	. . . . . . 7 |- ( ( x e. I /\ ( X .x. ( Y ` x ) ) e. _V ) -> ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) = ( X .x. ( Y ` x ) ) )
18	12, 13, 17	sylancl 694	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) = ( X .x. ( Y ` x ) ) )
19	11, 18	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( X .x. ( Y ` x ) ) )
20	19	neeq1d 2853	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. <-> ( X .x. ( Y ` x ) ) =/= .0. ) )
21	20	rabbidva 3188	. . 3 |- ( ph -> { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } = { x e. I | ( X .x. ( Y ` x ) ) =/= .0. } )
22		rrgsupp.r	. . . . . . 7 |- ( ph -> R e. Ring )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> R e. Ring )
24	2	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> X e. E )
25	7	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( Y ` x ) e. B )
26		rrgval.e	. . . . . . 7 |- E = (RLReg ` R )
27		rrgval.b	. . . . . . 7 |- B = ( Base ` R )
28		rrgval.t	. . . . . . 7 |- .x. = ( .r ` R )
29		rrgval.z	. . . . . . 7 |- .0. = ( 0g ` R )
30	26, 27, 28, 29	rrgeq0 19290	. . . . . 6 |- ( ( R e. Ring /\ X e. E /\ ( Y ` x ) e. B ) -> ( ( X .x. ( Y ` x ) ) = .0. <-> ( Y ` x ) = .0. ) )
31	23, 24, 25, 30	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) = .0. <-> ( Y ` x ) = .0. ) )
32	31	necon3bid 2838	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) =/= .0. <-> ( Y ` x ) =/= .0. ) )
33	32	rabbidva 3188	. . 3 |- ( ph -> { x e. I | ( X .x. ( Y ` x ) ) =/= .0. } = { x e. I | ( Y ` x ) =/= .0. } )
34	21, 33	eqtrd 2656	. 2 |- ( ph -> { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } = { x e. I | ( Y ` x ) =/= .0. } )
35		ovex 6678	. . . . . 6 |- ( X .x. ( Y ` y ) ) e. _V
36	35, 16	fnmpti 6022	. . . . 5 |- ( y e. I |-> ( X .x. ( Y ` y ) ) ) Fn I
37		fneq1 5979	. . . . 5 |- ( ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) -> ( ( ( I X. { X } ) oF .x. Y ) Fn I <-> ( y e. I |-> ( X .x. ( Y ` y ) ) ) Fn I ) )
38	36, 37	mpbiri 248	. . . 4 |- ( ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) -> ( ( I X. { X } ) oF .x. Y ) Fn I )
39	9, 38	syl 17	. . 3 |- ( ph -> ( ( I X. { X } ) oF .x. Y ) Fn I )
40		fvex 6201	. . . . 5 |- ( 0g ` R ) e. _V
41	29, 40	eqeltri 2697	. . . 4 |- .0. e. _V
42	41	a1i 11	. . 3 |- ( ph -> .0. e. _V )
43		suppvalfn 7302	. . 3 |- ( ( ( ( I X. { X } ) oF .x. Y ) Fn I /\ I e. V /\ .0. e. _V ) -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } )
44	39, 1, 42, 43	syl3anc 1326	. 2 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } )
45		ffn 6045	. . . 4 |- ( Y : I --> B -> Y Fn I )
46	7, 45	syl 17	. . 3 |- ( ph -> Y Fn I )
47		suppvalfn 7302	. . 3 |- ( ( Y Fn I /\ I e. V /\ .0. e. _V ) -> ( Y supp .0. ) = { x e. I | ( Y ` x ) =/= .0. } )
48	46, 1, 42, 47	syl3anc 1326	. 2 |- ( ph -> ( Y supp .0. ) = { x e. I | ( Y ` x ) =/= .0. } )
49	34, 44, 48	3eqtr4d 2666	1 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = ( Y supp .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Ringcrg 18547  RLRegcrlreg 19279

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ring 18549  df-rlreg 19283

This theorem is referenced by:  mdegvsca  23836  deg1mul3  23875