Theorem linc0scn0 49750

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 488	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod)
2		linc0scn0.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 21176	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2772	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6914	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc0scn0.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 20527	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc0scn0.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 20529	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 521	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
11	3, 10	syl 18	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
12	11	ad2antrr 739	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13		ifcl 4531	. . . . . 6 ⊢ (( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑍, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
14	12, 13	syl 18	. . . . 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 7141	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
17		fvex 6924	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝑀 ∈ LMod → (Base‘(Scalar‘𝑀)) ∈ V)
19		elmapg 8890	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
20	18, 19	sylan 592	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	16, 20	mpbird 260	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
22		linc0scn0.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
23	22	pweqi 4576	. . . . 5 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
24	23	eleq2i 2856	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
25	24	bilani 510	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
26		lincval 49736	. . 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))))
27	1, 21, 25, 26	syl3anc 1399	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
28		simpr 490	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
29	6	fvexi 6925	. . . . . . . 8 ⊢ 1 ∈ V
30	8	fvexi 6925	. . . . . . . 8 ⊢ 0 ∈ V
31	29, 30	ifex 4536	. . . . . . 7 ⊢ if(𝑣 = 𝑍, 1 , 0 ) ∈ V
32		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑍 ↔ 𝑣 = 𝑍))
33	32	ifbid 4509	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑍, 1 , 0 ) = if(𝑣 = 𝑍, 1 , 0 ))
34	33, 15	fvmptg 7017	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑍, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
35	28, 31, 34	sylancl 598	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 ))
36	35	oveq1d 7461	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣))
37		ovif 7544	. . . . . 6 ⊢ (if(𝑣 = 𝑍, 1 , 0 )( ·𝑠 ‘𝑀)𝑣) = if(𝑣 = 𝑍, ( 1 ( ·𝑠 ‘𝑀)𝑣), ( 0 ( ·𝑠 ‘𝑀)𝑣))
38	37	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)))
39		oveq2 7454	. . . . . . . 8 ⊢ (𝑣 = 𝑍 → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
40	39	adantl 487	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = ( 1 ( ·𝑠 ‘𝑀)𝑍))
41		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
42	2, 41, 6	lmod1cl 21197	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
43	42	ancli 558	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
44	43	adantr 486	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
45	44	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)))
46		eqid 2763	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
47		linc0scn0.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑀)
48	2, 46, 41, 47	lmodvs0 21204	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆)) → ( 1 ( ·𝑠 ‘𝑀)𝑍) = 𝑍)
49	45, 48	syl 18	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑍) = 𝑍)
50	40, 49	eqtrd 2798	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
51	1	adantr 486	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
52		elelpwi 4570	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
53	52	expcom 419	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
54	53	adantl 487	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
55	54	imp 412	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
56	22, 2, 46, 8, 47	lmod0vs 21203	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
57	51, 55, 56	syl2anc 596	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
58	57	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ ¬ 𝑣 = 𝑍) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = 𝑍)
59	50, 58	ifeqda 4522	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑍, ( 1 ( ·𝑠 ‘𝑀)𝑣), ( 0 ( ·𝑠 ‘𝑀)𝑣)) = 𝑍)
60	36, 38, 59	3eqtrd 2802	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = 𝑍)
61	60	mpteq2dva 5202	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍))
62	61	oveq2d 7462	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)))
63		lmodgrp 21175	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
64	63	grpmndd 19189	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
65	47	gsumz 19064	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
66	64, 65	sylan 592	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍)
67	27, 62, 66	3eqtrd 2802	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485  𝒫 cpw 4560   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ↑_m cmap 8878  Basecbs 17419  Scalarcsca 17463   ·_𝑠 cvsca 17464  0_gc0g 17642   Σ_g cgsu 17643  Mndcmnd 18955  1_rcur 20440  Ringcrg 20492  LModclmod 21168   linC clinc 49731

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-seq 14175  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-0g 17644  df-gsum 17645  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-grp 19179  df-minusg 19180  df-cmn 20029  df-abl 20030  df-mgp 20394  df-rng 20408  df-ur 20441  df-ring 20494  df-lmod 21170  df-linc 49733

This theorem is used by:  el0ldep  49793