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Theorem islindf 21367

Description: Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
islindf.b	⊢ 𝐵 = (Base‘𝑊)
islindf.v	⊢ · = ( ·_𝑠 ‘𝑊)
islindf.k	⊢ 𝐾 = (LSpan‘𝑊)
islindf.s	⊢ 𝑆 = (Scalar‘𝑊)
islindf.n	⊢ 𝑁 = (Base‘𝑆)
islindf.z	⊢ 0 = (0_g‘𝑆)

**Assertion**
Ref	Expression
islindf	⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))

Distinct variable groups: 𝑘,𝐹,𝑥 𝑘,𝑁 𝑘,𝑊,𝑥 0 ,𝑘 Allowed substitution hints: 𝐵(𝑥,𝑘) 𝑆(𝑥,𝑘) · (𝑥,𝑘) 𝐾(𝑥,𝑘) 𝑁(𝑥) 𝑋(𝑥,𝑘) 𝑌(𝑥,𝑘) 0 (𝑥)

Proof of Theorem islindf Dummy variables 𝑓 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 6699	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
2	1	adantr 482	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
3		dmeq 5904	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4	3	adantr 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → dom 𝑓 = dom 𝐹)
5		fveq2 6892	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
6		islindf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
7	5, 6	eqtr4di 2791	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
8	7	adantl 483	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (Base‘𝑤) = 𝐵)
9	4, 8	feq23d 6713	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝐹:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹⟶𝐵))
10	2, 9	bitrd 279	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹⟶𝐵))
11		fvex 6905	. . . . . 6 ⊢ (Scalar‘𝑤) ∈ V
12		fveq2 6892	. . . . . . . . 9 ⊢ (𝑠 = (Scalar‘𝑤) → (Base‘𝑠) = (Base‘(Scalar‘𝑤)))
13		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑠 = (Scalar‘𝑤) → (0_g‘𝑠) = (0_g‘(Scalar‘𝑤)))
14	13	sneqd 4641	. . . . . . . . 9 ⊢ (𝑠 = (Scalar‘𝑤) → {(0_g‘𝑠)} = {(0_g‘(Scalar‘𝑤))})
15	12, 14	difeq12d 4124	. . . . . . . 8 ⊢ (𝑠 = (Scalar‘𝑤) → ((Base‘𝑠) ∖ {(0_g‘𝑠)}) = ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}))
16	15	raleqdv 3326	. . . . . . 7 ⊢ (𝑠 = (Scalar‘𝑤) → (∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))))
17	16	ralbidv 3178	. . . . . 6 ⊢ (𝑠 = (Scalar‘𝑤) → (∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))))
18	11, 17	sbcie 3821	. . . . 5 ⊢ ([(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))
19		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
20		islindf.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑊)
21	19, 20	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑆)
22	21	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑆))
23		islindf.n	. . . . . . . . . 10 ⊢ 𝑁 = (Base‘𝑆)
24	22, 23	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝑁)
25	21	fveq2d 6896	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (0_g‘(Scalar‘𝑤)) = (0_g‘𝑆))
26		islindf.z	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑆)
27	25, 26	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (0_g‘(Scalar‘𝑤)) = 0 )
28	27	sneqd 4641	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → {(0_g‘(Scalar‘𝑤))} = { 0 })
29	24, 28	difeq12d 4124	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
30	29	adantl 483	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
31		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·_𝑠 ‘𝑤) = ( ·_𝑠 ‘𝑊))
32		islindf.v	. . . . . . . . . . . 12 ⊢ · = ( ·_𝑠 ‘𝑊)
33	31, 32	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·_𝑠 ‘𝑤) = · )
34	33	adantl 483	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ( ·_𝑠 ‘𝑤) = · )
35		eqidd 2734	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → 𝑘 = 𝑘)
36		fveq1 6891	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
37	36	adantr 482	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝑓‘𝑥) = (𝐹‘𝑥))
38	34, 35, 37	oveq123d 7430	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) = (𝑘 · (𝐹‘𝑥)))
39		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
40		islindf.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (LSpan‘𝑊)
41	39, 40	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝐾)
42	41	adantl 483	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (LSpan‘𝑤) = 𝐾)
43		imaeq1 6055	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝑓 ∖ {𝑥})))
44	3	difeq1d 4122	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∖ {𝑥}) = (dom 𝐹 ∖ {𝑥}))
45	44	imaeq2d 6060	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝐹 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
46	43, 45	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
47	46	adantr 482	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
48	42, 47	fveq12d 6899	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
49	38, 48	eleq12d 2828	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ((𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
50	49	notbid 318	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
51	30, 50	raleqbidv 3343	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
52	4, 51	raleqbidv 3343	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → (∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0_g‘(Scalar‘𝑤))}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
53	18, 52	bitrid 283	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ([(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
54	10, 53	anbi12d 632	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝑊) → ((𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
55		df-lindf 21361	. . 3 ⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
56	54, 55	brabga 5535	. 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
57	56	ancoms 460	1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 [wsbc 3778 ∖ cdif 3946 {csn 4629 class class class wbr 5149 dom cdm 5677 “ cima 5680 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 ·_𝑠 cvsca 17201 0_gc0g 17385 LSpanclspn 20582 LIndF clindf 21359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-lindf 21361

This theorem is referenced by: islinds2 21368 islindf2 21369 lindff 21370 lindfind 21371 f1lindf 21377 lsslindf 21385 lindfpropd 32498 matunitlindf 36486

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