Theorem linc0scn0 48416

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 482	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		linc0scn0.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 20781	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2739	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6864	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc0scn0.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 20181	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc0scn0.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 20183	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 511	. . . . . . . 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 726	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13		ifcl 4537	. . . . . 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 7089	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
17		fvex 6874	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
19		elmapg 8815	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
20	18, 19	sylan 580	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	16, 20	mpbird 257	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
22		linc0scn0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
23	22	pweqi 4582	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
24	23	eleq2i 2821	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
25	24	biimpi 216	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	adantl 481	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
27		lincval 48402	. . 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 1373	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
29		simpr 484	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
30	6	fvexi 6875	. . . . . . . 8 ⊢ 1 ∈ V
31	8	fvexi 6875	. . . . . . . 8 ⊢ 0 ∈ V
32	30, 31	ifex 4542	. . . . . . 7 ⊢ if(𝑣 = 𝑍, 1 , 0 ) ∈ V
33		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑍 ↔ 𝑣 = 𝑍))
34	33	ifbid 4515	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑍, 1 , 0 ) = if(𝑣 = 𝑍, 1 , 0 ))
35	34, 15	fvmptg 6969	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑍, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
36	29, 32, 35	sylancl 586	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
37	36	oveq1d 7405	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣))
38		ovif 7490	. . . . . 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 7398	. . . . . . . 8 ⊢ (𝑣 = 𝑍 → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
41	40	adantl 481	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
42		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
43	2, 42, 6	lmod1cl 20802	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
44	43	ancli 548	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
45	44	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
46	45	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
47		eqid 2730	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48		linc0scn0.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑀)
49	2, 47, 42, 48	lmodvs0 20809	. . . . . . . 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 2765	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
52	1	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
53		elelpwi 4576	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
54	53	expcom 413	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
56	55	imp 406	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
57	22, 2, 47, 8, 48	lmod0vs 20808	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
58	52, 56, 57	syl2anc 584	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
59	58	adantr 480	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ ¬ 𝑣 = 𝑍) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
60	51, 59	ifeqda 4528	. . . . 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 2769	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
62	61	mpteq2dva 5203	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
63	62	oveq2d 7406	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
64		lmodgrp 20780	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
65	64	grpmndd 18885	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
66	48	gsumz 18770	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
67	65, 66	sylan 580	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
68	28, 63, 67	3eqtrd 2769	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ifcif 4491  𝒫 cpw 4566   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  Basecbs 17186  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  1_rcur 20097  Ringcrg 20149  LModclmod 20773   linC clinc 48397

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-seq 13974  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-0g 17411  df-gsum 17412  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-lmod 20775  df-linc 48399

This theorem is referenced by:  el0ldep  48459