Theorem rrgsupp 14574

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 5843	. . . . . . . . 9 |- ( ( ph /\ y e. I ) -> ( Y ` y ) e. B )
6		fconstmpt 4822	. . . . . . . . . 10 |- ( I X. { X } ) = ( y e. I |-> X )
7	6	a1i 9	. . . . . . . . 9 |- ( ph -> ( I X. { X } ) = ( y e. I |-> X ) )
8	4	feqmptd 5756	. . . . . . . . 9 |- ( ph -> Y = ( y e. I |-> ( Y ` y ) ) )
9	1, 3, 5, 7, 8	offval2 6318	. . . . . . . 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 5697	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) )
12		eqid 2238	. . . . . . 7 |- ( y e. I |-> ( X .x. ( Y ` y ) ) ) = ( y e. I |-> ( X .x. ( Y ` y ) ) )
13		fveq2 5695	. . . . . . . 8 |- ( y = x -> ( Y ` y ) = ( Y ` x ) )
14	13	oveq2d 6101	. . . . . . 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 14571	. . . . . . . . . . 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 5843	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( Y ` x ) e. B )
27	19, 20	ringcl 14317	. . . . . . . 8 |- ( ( R e. Ring /\ X e. B /\ ( Y ` x ) e. B ) -> ( X .x. ( Y ` x ) ) e. B )
28	17, 25, 26, 27	syl3anc 1278	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( X .x. ( Y ` x ) ) e. B )
29	12, 14, 15, 28	fvmptd3 5799	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( y e. I |-> ( X .x. ( Y ` y ) ) ) ` x ) = ( X .x. ( Y ` x ) ) )
30	11, 29	eqtrd 2271	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( ( I X. { X } ) oF .x. Y ) ` x ) = ( X .x. ( Y ` x ) ) )
31	30	neeq1d 2438	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. <-> ( X .x. ( Y ` x ) ) =/= .0. ) )
32	31	rabbidva 2809	. . 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 14573	. . . . . 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 1278	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) = .0. <-> ( Y ` x ) = .0. ) )
36	35	necon3bid 2461	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( X .x. ( Y ` x ) ) =/= .0. <-> ( Y ` x ) =/= .0. ) )
37	36	rabbidva 2809	. . 3 |- ( ph -> { x e. I | ( X .x. ( Y ` x ) ) =/= .0. } = { x e. I | ( Y ` x ) =/= .0. } )
38	32, 37	eqtrd 2271	. 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 14317	. . . . . . 7 |- ( ( R e. Ring /\ X e. B /\ ( Y ` y ) e. B ) -> ( X .x. ( Y ` y ) ) e. B )
42	39, 40, 5, 41	syl3anc 1278	. . . . . 6 |- ( ( ph /\ y e. I ) -> ( X .x. ( Y ` y ) ) e. B )
43	42	ralrimiva 2623	. . . . 5 |- ( ph -> A. y e. I ( X .x. ( Y ` y ) ) e. B )
44	12	fnmpt 5510	. . . . 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 5471	. . . 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 14326	. . . 4 |- ( R e. Ring -> .0. e. B )
49	16, 48	syl 14	. . 3 |- ( ph -> .0. e. B )
50		suppvalfn 6481	. . 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 1278	. 2 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = { x e. I | ( ( ( I X. { X } ) oF .x. Y ) ` x ) =/= .0. } )
52	4	ffnd 5534	. . 3 |- ( ph -> Y Fn I )
53		suppvalfn 6481	. . 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 1278	. 2 |- ( ph -> ( Y supp .0. ) = { x e. I | ( Y ` x ) =/= .0. } )
55	38, 51, 54	3eqtr4d 2281	1 |- ( ph -> ( ( ( I X. { X } ) oF .x. Y ) supp .0. ) = ( Y supp .0. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   {csn 3709   |-> cmpt 4192   X. cxp 4772   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   oFcof 6300   supp csupp 6475   Basecbs 13352   .rcmulr 13432   0gc0g 13610   Ringcrg 14300  RLRegcrlreg 14563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-supp 6476  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-mgp 14218  df-ring 14302  df-rlreg 14566

This theorem is used by: (None)