Theorem rrgsupp 14344

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 u 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 276	. . . . . . . . 9 |- ( ( ph /\ y e. I ) -> X e. E )
4		rrgsupp.y	. . . . . . . . . 10 |- ( ph -> Y : I --> B )
5	4	ffvelcdmda 5790	. . . . . . . . 9 |- ( ( ph /\ y e. I ) -> ( Y ` y ) e. B )
6		fconstmpt 4779	. . . . . . . . . 10 |- ( I X. { X } ) = ( y e. I |-> X )
7	6	a1i 9	. . . . . . . . 9 |- ( ph -> ( I X. { X } ) = ( y e. I |-> X ) )
8	4	feqmptd 5708	. . . . . . . . 9 |- ( ph -> Y = ( y e. I |-> ( Y ` y ) ) )
9	1, 3, 5, 7, 8	offval2 6260	. . . . . . . 8 |- ( ph -> ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) )
10	9	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( I X. { X } ) oF .x. Y ) = ( y e. I |-> ( X .x. ( Y ` y ) ) ) )
11	10	fveq1d 5650	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) )
12		eqid 2231	. . . . . . 7 |- ( y e. I |-> ( X .x. ( Y ` y ) ) ) = ( y e. I |-> ( X .x. ( Y ` y ) ) )
13		fveq2 5648	. . . . . . . 8 |- ( y = x -> ( Y ` y ) = ( Y ` x ) )
14	13	oveq2d 6044	. . . . . . 7 |- ( y = x -> ( X .x. ( Y ` y ) ) = ( X .x. ( Y ` x ) ) )
15		simpr 110	. . . . . . 7 |- ( ( ph /\ x e. I ) -> x e. I )
16		rrgsupp.r	. . . . . . . . 9 |- ( ph -> R e. Ring )
17	16	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> R e. Ring )
18		rrgval.e	. . . . . . . . . . . 12 |- E = (RLReg ` R )
19		rrgval.b	. . . . . . . . . . . 12 |- B = ( Base ` R )
20		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
21		rrgval.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` R )
22	18, 19, 20, 21	isrrg 14341	. . . . . . . . . . 11 |- ( X e. E <-> ( X e. B /\ A. u e. B ( ( X .x. u ) = .0. -> u = .0. ) ) )
23	2, 22	sylib 122	. . . . . . . . . 10 |- ( ph -> ( X e. B /\ A. u e. B ( ( X .x. u ) = .0. -> u = .0. ) ) )
24	23	simpld 112	. . . . . . . . 9 |- ( ph -> X e. B )
25	24	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> X e. B )
26	4	ffvelcdmda 5790	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( Y ` x ) e. B )
27	19, 20	ringcl 14090	. . . . . . . 8 |- ( ( R e. Ring /\ X e. B /\ ( Y ` x ) e. B ) -> ( X .x. ( Y ` x ) ) e. B )
28	17, 25, 26, 27	syl3anc 1274	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( X .x. ( Y ` x ) ) e. B )
29	12, 14, 15, 28	fvmptd3 5749	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) = ( X .x. ( Y ` x ) ) )
30	11, 29	eqtrd 2264	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( X .x. ( Y ` x ) ) )
31	30	neeq1d 2421	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. <-> ( X .x. ( Y ` x ) ) =/= .0. ) )
32	31	rabbidva 2791	. . 3 |- ( ph -> { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } = { x e. I | ( X .x. ( Y ` x ) ) =/= .0. } )
33	2	adantr 276	. . . . . 6 |- ( ( ph /\ x e. I ) -> X e. E )
34	18, 19, 20, 21	rrgeq0 14343	. . . . . 6 |- ( ( R e. Ring /\ X e. E /\ ( Y ` x ) e. B ) -> ( ( X .x. ( Y ` x ) ) = .0. <-> ( Y ` x ) = .0. ) )
35	17, 33, 26, 34	syl3anc 1274	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) = .0. <-> ( Y ` x ) = .0. ) )
36	35	necon3bid 2444	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) =/= .0. <-> ( Y ` x ) =/= .0. ) )
37	36	rabbidva 2791	. . 3 |- ( ph -> { x e. I | ( X .x. ( Y ` x ) ) =/= .0. } = { x e. I | ( Y ` x ) =/= .0. } )
38	32, 37	eqtrd 2264	. 2 |- ( ph -> { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } = { x e. I | ( Y ` x ) =/= .0. } )
39	16	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> R e. Ring )
40	24	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. I ) -> X e. B )
41	19, 20	ringcl 14090	. . . . . . 7 |- ( ( R e. Ring /\ X e. B /\ ( Y ` y ) e. B ) -> ( X .x. ( Y ` y ) ) e. B )
42	39, 40, 5, 41	syl3anc 1274	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( X .x. ( Y ` y ) ) e. B )
43	42	ralrimiva 2606	. . . . 5 |- ( ph -> A. y e. I ( X .x. ( Y ` y ) ) e. B )
44	12	fnmpt 5466	. . . . 5 |- ( A. y e. I ( X .x. ( Y ` y ) ) e. B -> ( y e. I |-> ( X .x. ( Y ` y ) ) ) Fn I )
45	43, 44	syl 14	. . . 4 |- ( ph -> ( y e. I |-> ( X .x. ( Y ` y ) ) ) Fn I )
46	9	fneq1d 5427	. . . 4 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) Fn I <-> ( y e. I |-> ( X .x. ( Y ` y ) ) ) Fn I ) )
47	45, 46	mpbird 167	. . 3 |- ( ph -> ( ( I X. { X } ) oF .x. Y ) Fn I )
48	19, 21	ring0cl 14098	. . . 4 |- ( R e. Ring -> .0. e. B )
49	16, 48	syl 14	. . 3 |- ( ph -> .0. e. B )
50		suppvalfn 6419	. . 3 |- ( ( ( ( I X. { X } ) oF .x. Y ) Fn I /\ I e. V /\ .0. e. B ) -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } )
51	47, 1, 49, 50	syl3anc 1274	. 2 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } )
52	4	ffnd 5490	. . 3 |- ( ph -> Y Fn I )
53		suppvalfn 6419	. . 3 |- ( ( Y Fn I /\ I e. V /\ .0. e. B ) -> ( Y supp .0. ) = { x e. I | ( Y ` x ) =/= .0. } )
54	52, 1, 49, 53	syl3anc 1274	. 2 |- ( ph -> ( Y supp .0. ) = { x e. I | ( Y ` x ) =/= .0. } )
55	38, 51, 54	3eqtr4d 2274	1 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = ( Y supp .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   {crab 2515   {csn 3673   |-> cmpt 4155   X. cxp 4729   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   oFcof 6242   supp csupp 6413   Basecbs 13145   .rcmulr 13224   0gc0g 13402   Ringcrg 14073  RLRegcrlreg 14333

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-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  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-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-of 6244  df-supp 6414  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-mgp 13998  df-ring 14075  df-rlreg 14336

This theorem is referenced by: (None)