Theorem mndpsuppfi 18753

Description: The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)

Hypothesis

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

Assertion

Ref	Expression
mndpsuppfi	⊢ (((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)

Proof of Theorem mndpsuppfi

Step	Hyp	Ref	Expression
1		unfi 9194	. . 3 ⊢ (((𝐴 supp (0g‘𝑀)) ∈ Fin ∧ (𝐵 supp (0g‘𝑀)) ∈ Fin) → ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))) ∈ Fin)
2	1	3ad2ant3 1135	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ ((𝐴 supp (0g‘𝑀)) ∈ Fin ∧ (𝐵 supp (0g‘𝑀)) ∈ Fin)) → ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))) ∈ Fin)
3		mndpsuppfi.r	. . . 4 ⊢ 𝑅 = (Base‘𝑀)
4	3	mndpsuppss 18752	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))))
5	4	3adant3 1132	. 2 ⊢ (((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)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))))
6		ssfi 9196	. 2 ⊢ ((((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))) ∈ Fin ∧ ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀)))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ∈ Fin)
7	2, 5, 6	syl2anc 584	1 ⊢ (((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)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∪ cun 3931   ⊆ wss 3933  ‘cfv 6542  (class class class)co 7414   ∘_f cof 7678   supp csupp 8168   ↑_m cmap 8849  Fincfn 8968  Basecbs 17230  +_gcplusg 17277  0_gc0g 17460  Mndcmnd 18721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-om 7871  df-1st 7997  df-2nd 7998  df-supp 8169  df-1o 8489  df-map 8851  df-en 8969  df-fin 8972  df-0g 17462  df-mgm 18627  df-sgrp 18706  df-mnd 18722

This theorem is referenced by:  mndpfsupp  18754