Theorem lincvalsc0 42210

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 473	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		lincvalsc0.s	. . . . . . . 8 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	eqcomi 2631	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = 𝑆
4	3	fveq2i 6194	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
5		lincvalsc0.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑆)
6	2, 4, 5	lmod0cl 18889	. . . . . . 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 6385	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
11		fvexd 6203	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
12		elmapg 7870	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
13	11, 12	sylan 488	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
14	10, 13	mpbird 247	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
15		lincvalsc0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
16	15	pweqi 4162	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
17	16	eleq2i 2693	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
18	17	biimpi 206	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
19	18	adantl 482	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
20		lincval 42198	. . 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 1326	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
22		simpr 477	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
23		fvex 6201	. . . . . . . 8 ⊢ (0g‘𝑆) ∈ V
24	5, 23	eqeltri 2697	. . . . . . 7 ⊢ 0 ∈ V
25		eqidd 2623	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → 0 = 0 )
26	25, 9	fvmptg 6280	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 0 ∈ V) → (𝐹‘𝑣) = 0 )
27	22, 24, 26	sylancl 694	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = 0 )
28	27	oveq1d 6665	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
29	1	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
30		elelpwi 4171	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
31	30	expcom 451	. . . . . . . 8 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
32	31	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
33	32	imp 445	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
34		eqid 2622	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
35		lincvalsc0.z	. . . . . . 7 ⊢ 𝑍 = (0g‘𝑀)
36	15, 2, 34, 5, 35	lmod0vs 18896	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
37	29, 33, 36	syl2anc 693	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
38	28, 37	eqtrd 2656	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
39	38	mpteq2dva 4744	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
40	39	oveq2d 6666	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
41		lmodgrp 18870	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
42		grpmnd 17429	. . . 4 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
43	41, 42	syl 17	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
44	35	gsumz 17374	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
45	43, 44	sylan 488	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
46	21, 40, 45	3eqtrd 2660	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294  Grpcgrp 17422  LModclmod 18863   linC clinc 42193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-seq 12802  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865  df-linc 42195

This theorem is referenced by:  lcoc0  42211