Theorem lcomfsupp 19676

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	⊢ · = ( ·𝑠 ‘𝑊)
lcomf.b	⊢ 𝐵 = (Base‘𝑊)
lcomf.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lcomf.g	⊢ (𝜑 → 𝐺:𝐼⟶𝐾)
lcomf.h	⊢ (𝜑 → 𝐻:𝐼⟶𝐵)
lcomf.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
lcomfsupp.z	⊢ 0 = (0g‘𝑊)
lcomfsupp.y	⊢ 𝑌 = (0g‘𝐹)
lcomfsupp.j	⊢ (𝜑 → 𝐺 finSupp 𝑌)

Assertion

Ref	Expression
lcomfsupp	⊢ (𝜑 → (𝐺 ∘f · 𝐻) finSupp 0 )

Proof of Theorem lcomfsupp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcomfsupp.j	. . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑌)
2	1	fsuppimpd 8842	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑌) ∈ Fin)
3		lcomf.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4		lcomf.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
5		lcomf.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
6		lcomf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
7		lcomf.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lcomf.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐼⟶𝐾)
9		lcomf.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝐼⟶𝐵)
10		lcomf.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
11	3, 4, 5, 6, 7, 8, 9, 10	lcomf 19675	. . . 4 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵)
12		eldifi 4105	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌)) → 𝑥 ∈ 𝐼)
13	8	ffnd 6517	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝐼)
14	13	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 Fn 𝐼)
15	9	ffnd 6517	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐼)
16	15	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 Fn 𝐼)
17	10	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉)
18		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
19		fnfvof 7425	. . . . . . 7 ⊢ (((𝐺 Fn 𝐼 ∧ 𝐻 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥)))
20	14, 16, 17, 18, 19	syl22anc 836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥)))
21	12, 20	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥)))
22		ssidd 3992	. . . . . . 7 ⊢ (𝜑 → (𝐺 supp 𝑌) ⊆ (𝐺 supp 𝑌))
23		lcomfsupp.y	. . . . . . . . 9 ⊢ 𝑌 = (0g‘𝐹)
24	23	fvexi 6686	. . . . . . . 8 ⊢ 𝑌 ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
26	8, 22, 10, 25	suppssr 7863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝐺‘𝑥) = 𝑌)
27	26	oveq1d 7173	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺‘𝑥) · (𝐻‘𝑥)) = (𝑌 · (𝐻‘𝑥)))
28	9	ffvelrnda 6853	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ 𝐵)
29		lcomfsupp.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑊)
30	6, 3, 5, 23, 29	lmod0vs 19669	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝐻‘𝑥) ∈ 𝐵) → (𝑌 · (𝐻‘𝑥)) = 0 )
31	7, 28, 30	syl2an2r 683	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑌 · (𝐻‘𝑥)) = 0 )
32	12, 31	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝑌 · (𝐻‘𝑥)) = 0 )
33	21, 27, 32	3eqtrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = 0 )
34	11, 33	suppss 7862	. . 3 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ⊆ (𝐺 supp 𝑌))
35	2, 34	ssfid 8743	. 2 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin)
36		inidm 4197	. . . . 5 ⊢ (𝐼 ∩ 𝐼) = 𝐼
37	13, 15, 10, 10, 36	offn 7422	. . . 4 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) Fn 𝐼)
38		fnfun 6455	. . . 4 ⊢ ((𝐺 ∘f · 𝐻) Fn 𝐼 → Fun (𝐺 ∘f · 𝐻))
39	37, 38	syl 17	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘f · 𝐻))
40		ovexd 7193	. . 3 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) ∈ V)
41	29	fvexi 6686	. . . 4 ⊢ 0 ∈ V
42	41	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
43		funisfsupp 8840	. . 3 ⊢ ((Fun (𝐺 ∘f · 𝐻) ∧ (𝐺 ∘f · 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐺 ∘f · 𝐻) finSupp 0 ↔ ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin))
44	39, 40, 42, 43	syl3anc 1367	. 2 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) finSupp 0 ↔ ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin))
45	35, 44	mpbird 259	1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∖ cdif 3935   class class class wbr 5068  Fun wfun 6351   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409   supp csupp 7832  Fincfn 8511   finSupp cfsupp 8835  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715  LModclmod 19636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-supp 7833  df-er 8291  df-en 8512  df-fin 8515  df-fsupp 8836  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-ring 19301  df-lmod 19638

This theorem is referenced by:  islindf4  20984  fedgmullem2  31028