Theorem lcomfsupp 18903

Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by AV, 15-Jul-2019.)

Hypotheses

Ref	Expression
lcomf.f	|- F = (Scalar ` W )
lcomf.k	|- K = ( Base ` F )
lcomf.s	|- .x. = ( .s ` W )
lcomf.b	|- B = ( Base ` W )
lcomf.w	|- ( ph -> W e. LMod )
lcomf.g	|- ( ph -> G : I --> K )
lcomf.h	|- ( ph -> H : I --> B )
lcomf.i	|- ( ph -> I e. V )
lcomfsupp.z	|- .0. = ( 0g ` W )
lcomfsupp.y	|- Y = ( 0g ` F )
lcomfsupp.j	|- ( ph -> G finSupp Y )

Assertion

Ref	Expression
lcomfsupp	|- ( ph -> ( G oF .x. H ) finSupp .0. )

Proof of Theorem lcomfsupp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcomfsupp.j	. . . 4 |- ( ph -> G finSupp Y )
2	1	fsuppimpd 8282	. . 3 |- ( ph -> ( G supp Y ) e. Fin )
3		lcomf.f	. . . . 5 |- F = (Scalar ` W )
4		lcomf.k	. . . . 5 |- K = ( Base ` F )
5		lcomf.s	. . . . 5 |- .x. = ( .s ` W )
6		lcomf.b	. . . . 5 |- B = ( Base ` W )
7		lcomf.w	. . . . 5 |- ( ph -> W e. LMod )
8		lcomf.g	. . . . 5 |- ( ph -> G : I --> K )
9		lcomf.h	. . . . 5 |- ( ph -> H : I --> B )
10		lcomf.i	. . . . 5 |- ( ph -> I e. V )
11	3, 4, 5, 6, 7, 8, 9, 10	lcomf 18902	. . . 4 |- ( ph -> ( G oF .x. H ) : I --> B )
12		eldifi 3732	. . . . . 6 |- ( x e. ( I \ ( G supp Y ) ) -> x e. I )
13		ffn 6045	. . . . . . . . 9 |- ( G : I --> K -> G Fn I )
14	8, 13	syl 17	. . . . . . . 8 |- ( ph -> G Fn I )
15	14	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> G Fn I )
16		ffn 6045	. . . . . . . . 9 |- ( H : I --> B -> H Fn I )
17	9, 16	syl 17	. . . . . . . 8 |- ( ph -> H Fn I )
18	17	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> H Fn I )
19	10	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> I e. V )
20		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. I ) -> x e. I )
21		fnfvof 6911	. . . . . . 7 |- ( ( ( G Fn I /\ H Fn I ) /\ ( I e. V /\ x e. I ) ) -> ( ( G oF .x. H ) ` x ) = ( ( G ` x ) .x. ( H ` x ) ) )
22	15, 18, 19, 20, 21	syl22anc 1327	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( ( G oF .x. H ) ` x ) = ( ( G ` x ) .x. ( H ` x ) ) )
23	12, 22	sylan2 491	. . . . 5 |- ( ( ph /\ x e. ( I \ ( G supp Y ) ) ) -> ( ( G oF .x. H ) ` x ) = ( ( G ` x ) .x. ( H ` x ) ) )
24		ssid 3624	. . . . . . . 8 |- ( G supp Y ) C_ ( G supp Y )
25	24	a1i 11	. . . . . . 7 |- ( ph -> ( G supp Y ) C_ ( G supp Y ) )
26		lcomfsupp.y	. . . . . . . . 9 |- Y = ( 0g ` F )
27		fvex 6201	. . . . . . . . 9 |- ( 0g ` F ) e. _V
28	26, 27	eqeltri 2697	. . . . . . . 8 |- Y e. _V
29	28	a1i 11	. . . . . . 7 |- ( ph -> Y e. _V )
30	8, 25, 10, 29	suppssr 7326	. . . . . 6 |- ( ( ph /\ x e. ( I \ ( G supp Y ) ) ) -> ( G ` x ) = Y )
31	30	oveq1d 6665	. . . . 5 |- ( ( ph /\ x e. ( I \ ( G supp Y ) ) ) -> ( ( G ` x ) .x. ( H ` x ) ) = ( Y .x. ( H ` x ) ) )
32	7	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. I ) -> W e. LMod )
33	9	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( H ` x ) e. B )
34		lcomfsupp.z	. . . . . . . 8 |- .0. = ( 0g ` W )
35	6, 3, 5, 26, 34	lmod0vs 18896	. . . . . . 7 |- ( ( W e. LMod /\ ( H ` x ) e. B ) -> ( Y .x. ( H ` x ) ) = .0. )
36	32, 33, 35	syl2anc 693	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( Y .x. ( H ` x ) ) = .0. )
37	12, 36	sylan2 491	. . . . 5 |- ( ( ph /\ x e. ( I \ ( G supp Y ) ) ) -> ( Y .x. ( H ` x ) ) = .0. )
38	23, 31, 37	3eqtrd 2660	. . . 4 |- ( ( ph /\ x e. ( I \ ( G supp Y ) ) ) -> ( ( G oF .x. H ) ` x ) = .0. )
39	11, 38	suppss 7325	. . 3 |- ( ph -> ( ( G oF .x. H ) supp .0. ) C_ ( G supp Y ) )
40		ssfi 8180	. . 3 |- ( ( ( G supp Y ) e. Fin /\ ( ( G oF .x. H ) supp .0. ) C_ ( G supp Y ) ) -> ( ( G oF .x. H ) supp .0. ) e. Fin )
41	2, 39, 40	syl2anc 693	. 2 |- ( ph -> ( ( G oF .x. H ) supp .0. ) e. Fin )
42		inidm 3822	. . . . 5 |- ( I i^i I ) = I
43	14, 17, 10, 10, 42	offn 6908	. . . 4 |- ( ph -> ( G oF .x. H ) Fn I )
44		fnfun 5988	. . . 4 |- ( ( G oF .x. H ) Fn I -> Fun ( G oF .x. H ) )
45	43, 44	syl 17	. . 3 |- ( ph -> Fun ( G oF .x. H ) )
46		ovexd 6680	. . 3 |- ( ph -> ( G oF .x. H ) e. _V )
47		fvex 6201	. . . . 5 |- ( 0g ` W ) e. _V
48	34, 47	eqeltri 2697	. . . 4 |- .0. e. _V
49	48	a1i 11	. . 3 |- ( ph -> .0. e. _V )
50		funisfsupp 8280	. . 3 |- ( ( Fun ( G oF .x. H ) /\ ( G oF .x. H ) e. _V /\ .0. e. _V ) -> ( ( G oF .x. H ) finSupp .0. <-> ( ( G oF .x. H ) supp .0. ) e. Fin ) )
51	45, 46, 49, 50	syl3anc 1326	. 2 |- ( ph -> ( ( G oF .x. H ) finSupp .0. <-> ( ( G oF .x. H ) supp .0. ) e. Fin ) )
52	41, 51	mpbird 247	1 |- ( ph -> ( G oF .x. H ) finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863

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-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-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-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-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865

This theorem is referenced by:  islindf4  20177