Theorem lincvalsc0 46113

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 483	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		lincvalsc0.s	. . . . . . . 8 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	eqcomi 2745	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = 𝑆
4	3	fveq2i 6828	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
5		lincvalsc0.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑆)
6	2, 4, 5	lmod0cl 20255	. . . . . . 7 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘(Scalar‘𝑀)))
7	6	adantr 481	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 0 ∈ (Base‘(Scalar‘𝑀)))
8	7	adantr 481	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ (Base‘(Scalar‘𝑀)))
9		lincvalsc0.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 )
10	8, 9	fmptd 7044	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
11		fvexd 6840	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
12		elmapg 8699	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
13	11, 12	sylan 580	. . . 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 4563	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
17	16	eleq2i 2828	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
18	17	biimpi 215	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
19	18	adantl 482	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
20		lincval 46101	. . 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 1370	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
22		simpr 485	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
23	5	fvexi 6839	. . . . . . 7 ⊢ 0 ∈ V
24		eqidd 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → 0 = 0 )
25	24, 9	fvmptg 6929	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 0 ∈ V) → (𝐹‘𝑣) = 0 )
26	22, 23, 25	sylancl 586	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = 0 )
27	26	oveq1d 7352	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
28	1	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
29		elelpwi 4557	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
30	29	expcom 414	. . . . . . . 8 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
31	30	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
32	31	imp 407	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
33		eqid 2736	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
34		lincvalsc0.z	. . . . . . 7 ⊢ 𝑍 = (0g‘𝑀)
35	15, 2, 33, 5, 34	lmod0vs 20262	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
36	28, 32, 35	syl2anc 584	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
37	27, 36	eqtrd 2776	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
38	37	mpteq2dva 5192	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
39	38	oveq2d 7353	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
40		lmodgrp 20236	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
41	40	grpmndd 18685	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
42	34	gsumz 18571	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
43	41, 42	sylan 580	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
44	21, 39, 43	3eqtrd 2780	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441  𝒫 cpw 4547   ↦ cmpt 5175  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337   ↑_m cmap 8686  Basecbs 17009  Scalarcsca 17062   ·_𝑠 cvsca 17063  0_gc0g 17247   Σ_g cgsu 17248  Mndcmnd 18482  LModclmod 20229   linC clinc 46096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-map 8688  df-seq 13823  df-0g 17249  df-gsum 17250  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-ring 19880  df-lmod 20231  df-linc 46098

This theorem is referenced by:  lcoc0  46114