Theorem mndpfsupp 18670

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 8815	. . . . 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 1134	. . 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 8815	. . . . 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 1134	. . 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 1199	. . 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 7647	. 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 9296	. . . 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 9296	. . . 4 ⊢ (𝐵 finSupp (0g‘𝑀) → (𝐵 supp (0g‘𝑀)) ∈ Fin)
13	10, 12	anim12i 613	. . 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 18669	. . 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 1165	. 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 7402	. . 3 ⊢ (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V
18		fvexd 6855	. . 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 9292	. . 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 587	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444   class class class wbr 5102  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631   supp csupp 8116   ↑_m cmap 8776  Fincfn 8895   finSupp cfsupp 9288  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Mndcmnd 18637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-1o 8411  df-map 8778  df-en 8896  df-fin 8899  df-fsupp 9289  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638

This theorem is referenced by:  elrgspnlem1  33166  lincsumcl  48393