Theorem mndpsuppfi 18675

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 9112	. . 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 18674	. . 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 9114	. 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 1540   ∈ wcel 2109   ∪ cun 3909   ⊆ wss 3911  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631   supp csupp 8116   ↑_m cmap 8776  Fincfn 8895  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Mndcmnd 18643

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-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644

This theorem is referenced by:  mndpfsupp  18676