Theorem lincvalsc0 45650

Description: The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)

Hypotheses

Ref	Expression
lincvalsc0.b	⊢ 𝐵 = (Base‘𝑀)
lincvalsc0.s	⊢ 𝑆 = (Scalar‘𝑀)
lincvalsc0.0	⊢ 0 = (0g‘𝑆)
lincvalsc0.z	⊢ 𝑍 = (0g‘𝑀)
lincvalsc0.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 )

Assertion

Ref	Expression
lincvalsc0	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Distinct variable groups:   𝑥,𝐵   𝑥,𝑀   𝑥,𝑉   𝑥, 0

Allowed substitution hints:   𝑆(𝑥)   𝐹(𝑥)   𝑍(𝑥)

Proof of Theorem lincvalsc0

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		lincvalsc0.s	. . . . . . . 8 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	eqcomi 2747	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = 𝑆
4	3	fveq2i 6759	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
5		lincvalsc0.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑆)
6	2, 4, 5	lmod0cl 20064	. . . . . . 7 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘(Scalar‘𝑀)))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 0 ∈ (Base‘(Scalar‘𝑀)))
8	7	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ (Base‘(Scalar‘𝑀)))
9		lincvalsc0.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 )
10	8, 9	fmptd 6970	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
11		fvexd 6771	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
12		elmapg 8586	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
13	11, 12	sylan 579	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
14	10, 13	mpbird 256	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
15		lincvalsc0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
16	15	pweqi 4548	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
17	16	eleq2i 2830	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
18	17	biimpi 215	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
19	18	adantl 481	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
20		lincval 45638	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
21	1, 14, 19, 20	syl3anc 1369	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
22		simpr 484	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
23	5	fvexi 6770	. . . . . . 7 ⊢ 0 ∈ V
24		eqidd 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → 0 = 0 )
25	24, 9	fvmptg 6855	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 0 ∈ V) → (𝐹‘𝑣) = 0 )
26	22, 23, 25	sylancl 585	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = 0 )
27	26	oveq1d 7270	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
28	1	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
29		elelpwi 4542	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
30	29	expcom 413	. . . . . . . 8 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
31	30	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
32	31	imp 406	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
33		eqid 2738	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
34		lincvalsc0.z	. . . . . . 7 ⊢ 𝑍 = (0g‘𝑀)
35	15, 2, 33, 5, 34	lmod0vs 20071	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
36	28, 32, 35	syl2anc 583	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
37	27, 36	eqtrd 2778	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
38	37	mpteq2dva 5170	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
39	38	oveq2d 7271	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
40		lmodgrp 20045	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
41	40	grpmndd 18504	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
42	34	gsumz 18389	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
43	41, 42	sylan 579	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
44	21, 39, 43	3eqtrd 2782	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  𝒫 cpw 4530   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  Mndcmnd 18300  LModclmod 20038   linC clinc 45633

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-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-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  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-pred 6191  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-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-map 8575  df-seq 13650  df-0g 17069  df-gsum 17070  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-ring 19700  df-lmod 20040  df-linc 45635

This theorem is referenced by:  lcoc0  45651