Theorem linc0scn0 48152

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 20888	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2749	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6923	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc0scn0.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 20289	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc0scn0.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 20290	. . . . . . . . 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 725	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13		ifcl 4593	. . . . . 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 7148	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
17		fvex 6933	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
19		elmapg 8897	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
20	18, 19	sylan 579	. . . 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 4638	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
24	23	eleq2i 2836	. . . . 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 48138	. . 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 1371	. 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 6934	. . . . . . . 8 ⊢ 1 ∈ V
31	8	fvexi 6934	. . . . . . . 8 ⊢ 0 ∈ V
32	30, 31	ifex 4598	. . . . . . 7 ⊢ if(𝑣 = 𝑍, 1 , 0 ) ∈ V
33		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑍 ↔ 𝑣 = 𝑍))
34	33	ifbid 4571	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑍, 1 , 0 ) = if(𝑣 = 𝑍, 1 , 0 ))
35	34, 15	fvmptg 7027	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑍, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
36	29, 32, 35	sylancl 585	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
37	36	oveq1d 7463	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣))
38		ovif 7548	. . . . . 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 7456	. . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
43	2, 42, 6	lmod1cl 20909	. . . . . . . . . . 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 725	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
47		eqid 2740	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48		linc0scn0.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑀)
49	2, 47, 42, 48	lmodvs0 20916	. . . . . . . 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 2780	. . . . . 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 4632	. . . . . . . . . . 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 20915	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
58	52, 56, 57	syl2anc 583	. . . . . . 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 4584	. . . . 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 2784	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
62	61	mpteq2dva 5266	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
63	62	oveq2d 7464	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
64		lmodgrp 20887	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
65	64	grpmndd 18986	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
66	48	gsumz 18871	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
67	65, 66	sylan 579	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
68	28, 63, 67	3eqtrd 2784	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ifcif 4548  𝒫 cpw 4622   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499   Σ_g cgsu 17500  Mndcmnd 18772  1_rcur 20208  Ringcrg 20260  LModclmod 20880   linC clinc 48133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-seq 14053  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-lmod 20882  df-linc 48135

This theorem is referenced by:  el0ldep  48195