Theorem mndpfsupp 18801

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 8846	. . . . 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 1147	. . 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 8846	. . . . 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 1147	. . 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 1212	. . 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 7674	. 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 9315	. . . 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 9315	. . . 4 ⊢ (𝐵 finSupp (0g‘𝑀) → (𝐵 supp (0g‘𝑀)) ∈ Fin)
13	10, 12	anim12i 622	. . 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 18800	. . 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 1178	. 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 7429	. . 3 ⊢ (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V
18		fvexd 6882	. . 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 9311	. . 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 596	. 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 723	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 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ∘_f cof 7658   supp csupp 8140   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9307  Basecbs 17245  +_gcplusg 17286  0_gc0g 17468  Mndcmnd 18768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-1o 8437  df-map 8810  df-en 8928  df-fin 8931  df-fsupp 9308  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769

This theorem is referenced by:  elrgspnlem1  33420  lincsumcl  49050