Theorem linc0scn0 44485

Description: If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)

Hypotheses

Ref	Expression
linc0scn0.b	⊢ 𝐵 = (Base‘𝑀)
linc0scn0.s	⊢ 𝑆 = (Scalar‘𝑀)
linc0scn0.0	⊢ 0 = (0g‘𝑆)
linc0scn0.1	⊢ 1 = (1r‘𝑆)
linc0scn0.z	⊢ 𝑍 = (0g‘𝑀)
linc0scn0.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))

Assertion

Ref	Expression
linc0scn0	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Distinct variable groups:   𝑥,𝐵   𝑥,𝑀   𝑥,𝑉   𝑥,𝑍   𝑥, 0   𝑥, 1

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

Proof of Theorem linc0scn0

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 485	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		linc0scn0.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 19644	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2832	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6675	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc0scn0.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 19320	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc0scn0.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 19321	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 514	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
11	3, 10	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
12	11	ad2antrr 724	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13		ifcl 4513	. . . . . 6 ⊢ (( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑍, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
14	12, 13	syl 17	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → if(𝑥 = 𝑍, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
15		linc0scn0.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))
16	14, 15	fmptd 6880	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
17		fvex 6685	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
19		elmapg 8421	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
20	18, 19	sylan 582	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	16, 20	mpbird 259	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
22		linc0scn0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
23	22	pweqi 4559	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
24	23	eleq2i 2906	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
25	24	biimpi 218	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	adantl 484	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
27		lincval 44471	. . 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))))
28	1, 21, 26, 27	syl3anc 1367	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
29		simpr 487	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
30	6	fvexi 6686	. . . . . . . 8 ⊢ 1 ∈ V
31	8	fvexi 6686	. . . . . . . 8 ⊢ 0 ∈ V
32	30, 31	ifex 4517	. . . . . . 7 ⊢ if(𝑣 = 𝑍, 1 , 0 ) ∈ V
33		eqeq1 2827	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑍 ↔ 𝑣 = 𝑍))
34	33	ifbid 4491	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑍, 1 , 0 ) = if(𝑣 = 𝑍, 1 , 0 ))
35	34, 15	fvmptg 6768	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑍, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
36	29, 32, 35	sylancl 588	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
37	36	oveq1d 7173	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣))
38		ovif 7253	. . . . . 6 ⊢ (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣) = if(𝑣 = 𝑍, ( 1 ( ·𝑠 ‘𝑀)𝑣), ( 0 ( ·𝑠 ‘𝑀)𝑣))
39	38	a1i 11	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣) = if(𝑣 = 𝑍, ( 1 ( ·𝑠 ‘𝑀)𝑣), ( 0 ( ·𝑠 ‘𝑀)𝑣)))
40		oveq2 7166	. . . . . . . 8 ⊢ (𝑣 = 𝑍 → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
41	40	adantl 484	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
42		eqid 2823	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
43	2, 42, 6	lmod1cl 19663	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
44	43	ancli 551	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
45	44	adantr 483	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
46	45	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
47		eqid 2823	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48		linc0scn0.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑀)
49	2, 47, 42, 48	lmodvs0 19670	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)) → ( 1 ( ·𝑠 ‘𝑀)𝑍) = 𝑍)
50	46, 49	syl 17	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑍) = 𝑍)
51	41, 50	eqtrd 2858	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
52	1	adantr 483	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
53		elelpwi 4553	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
54	53	expcom 416	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
55	54	adantl 484	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
56	55	imp 409	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
57	22, 2, 47, 8, 48	lmod0vs 19669	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
58	52, 56, 57	syl2anc 586	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
59	58	adantr 483	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ ¬ 𝑣 = 𝑍) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
60	51, 59	ifeqda 4504	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑍, ( 1 ( ·𝑠 ‘𝑀)𝑣), ( 0 ( ·𝑠 ‘𝑀)𝑣)) = 𝑍)
61	37, 39, 60	3eqtrd 2862	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
62	61	mpteq2dva 5163	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
63	62	oveq2d 7174	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
64		lmodgrp 19643	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
65		grpmnd 18112	. . . 4 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
66	64, 65	syl 17	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
67	48	gsumz 18002	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
68	66, 67	sylan 582	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
69	28, 63, 68	3eqtrd 2862	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ifcif 4469  𝒫 cpw 4541   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ↑_m cmap 8408  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715   Σ_g cgsu 16716  Mndcmnd 17913  Grpcgrp 18105  1_rcur 19253  Ringcrg 19299  LModclmod 19636   linC clinc 44466

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-seq 13373  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-plusg 16580  df-0g 16717  df-gsum 16718  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-mgp 19242  df-ur 19254  df-ring 19301  df-lmod 19638  df-linc 44468

This theorem is referenced by:  el0ldep  44528