Theorem mndpfsupp 45385

Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)

Hypothesis

Ref	Expression
mndpsuppfi.r	⊢ 𝑅 = (Base‘𝑀)

Assertion

Ref	Expression
mndpfsupp	⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → (𝐴 ∘f (+g‘𝑀)𝐵) finSupp (0g‘𝑀))

Proof of Theorem mndpfsupp

Step	Hyp	Ref	Expression
1		elmapfn 8546	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 Fn 𝑉)
2	1	adantr 484	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴 Fn 𝑉)
3	2	3ad2ant2 1136	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → 𝐴 Fn 𝑉)
4		elmapfn 8546	. . . . 5 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 Fn 𝑉)
5	4	adantl 485	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵 Fn 𝑉)
6	5	3ad2ant2 1136	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → 𝐵 Fn 𝑉)
7		simp1r 1200	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → 𝑉 ∈ 𝑋)
8	3, 6, 7, 7	offun 7482	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → Fun (𝐴 ∘f (+g‘𝑀)𝐵))
9		id 22	. . . . 5 ⊢ (𝐴 finSupp (0g‘𝑀) → 𝐴 finSupp (0g‘𝑀))
10	9	fsuppimpd 8992	. . . 4 ⊢ (𝐴 finSupp (0g‘𝑀) → (𝐴 supp (0g‘𝑀)) ∈ Fin)
11		id 22	. . . . 5 ⊢ (𝐵 finSupp (0g‘𝑀) → 𝐵 finSupp (0g‘𝑀))
12	11	fsuppimpd 8992	. . . 4 ⊢ (𝐵 finSupp (0g‘𝑀) → (𝐵 supp (0g‘𝑀)) ∈ Fin)
13	10, 12	anim12i 616	. . 3 ⊢ ((𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀)) → ((𝐴 supp (0g‘𝑀)) ∈ Fin ∧ (𝐵 supp (0g‘𝑀)) ∈ Fin))
14		mndpsuppfi.r	. . . 4 ⊢ 𝑅 = (Base‘𝑀)
15	14	mndpsuppfi 45384	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ ((𝐴 supp (0g‘𝑀)) ∈ Fin ∧ (𝐵 supp (0g‘𝑀)) ∈ Fin)) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)
16	13, 15	syl3an3 1167	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)
17		ovex 7246	. . 3 ⊢ (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V
18		fvexd 6732	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → (0g‘𝑀) ∈ V)
19		isfsupp 8989	. . 3 ⊢ (((𝐴 ∘f (+g‘𝑀)𝐵) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝐴 ∘f (+g‘𝑀)𝐵) finSupp (0g‘𝑀) ↔ (Fun (𝐴 ∘f (+g‘𝑀)𝐵) ∧ ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)))
20	17, 18, 19	sylancr 590	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → ((𝐴 ∘f (+g‘𝑀)𝐵) finSupp (0g‘𝑀) ↔ (Fun (𝐴 ∘f (+g‘𝑀)𝐵) ∧ ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)))
21	8, 16, 20	mpbir2and 713	1 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑀) ∧ 𝐵 finSupp (0g‘𝑀))) → (𝐴 ∘f (+g‘𝑀)𝐵) finSupp (0g‘𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3408   class class class wbr 5053  Fun wfun 6374   Fn wfn 6375  ‘cfv 6380  (class class class)co 7213   ∘_f cof 7467   supp csupp 7903   ↑_m cmap 8508  Fincfn 8626   finSupp cfsupp 8985  Basecbs 16760  +_gcplusg 16802  0_gc0g 16944  Mndcmnd 18173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-1o 8202  df-map 8510  df-en 8627  df-fin 8630  df-fsupp 8986  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174

This theorem is referenced by:  lincsumcl  45445