Theorem mndpfsupp 45600

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 8611	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 Fn 𝑉)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴 Fn 𝑉)
3	2	3ad2ant2 1132	. . 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 8611	. . . . 5 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 Fn 𝑉)
5	4	adantl 481	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵 Fn 𝑉)
6	5	3ad2ant2 1132	. . 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 1196	. . 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 7525	. 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 9065	. . . 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 9065	. . . 4 ⊢ (𝐵 finSupp (0g‘𝑀) → (𝐵 supp (0g‘𝑀)) ∈ Fin)
13	10, 12	anim12i 612	. . 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 45599	. . 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 1163	. 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 7288	. . 3 ⊢ (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V
18		fvexd 6771	. . 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 9062	. . 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 586	. 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 709	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509   supp csupp 7948   ↑_m cmap 8573  Fincfn 8691   finSupp cfsupp 9058  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Mndcmnd 18300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-1o 8267  df-map 8575  df-en 8692  df-fin 8695  df-fsupp 9059  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301

This theorem is referenced by:  lincsumcl  45660