lindfpropd - Mathbox for Thierry Arnoux

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Theorem lindfpropd 32498

Description: Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.)

**Hypotheses**
Ref	Expression
lindfpropd.1	⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
lindfpropd.2	⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
lindfpropd.3	⊢ (𝜑 → (0_g‘(Scalar‘𝐾)) = (0_g‘(Scalar‘𝐿)))
lindfpropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
lindfpropd.5	⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
lindfpropd.6	⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
lindfpropd.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lindfpropd.l	⊢ (𝜑 → 𝐿 ∈ 𝑊)
lindfpropd.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

**Assertion**
Ref	Expression
lindfpropd	⊢ (𝜑 → (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿))

Distinct variable groups: 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝑥,𝑋,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝑉(𝑥,𝑦) 𝑊(𝑥,𝑦)

Proof of Theorem lindfpropd Dummy variables 𝑖 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lindfpropd.2	. . . . . . . 8 ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
2		lindfpropd.3	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘(Scalar‘𝐾)) = (0_g‘(Scalar‘𝐿)))
3	2	sneqd 4641	. . . . . . . 8 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐾))} = {(0_g‘(Scalar‘𝐿))})
4	1, 3	difeq12d 4124	. . . . . . 7 ⊢ (𝜑 → ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}))
5	4	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}))
6		simplll 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → 𝜑)
7		simpr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}))
8	7	eldifad 3961	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → 𝑘 ∈ (Base‘(Scalar‘𝐾)))
9		simpr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) → 𝑋:dom 𝑋⟶(Base‘𝐾))
10	9	ffvelcdmda 7087	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (𝑋‘𝑖) ∈ (Base‘𝐾))
11	10	adantr 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → (𝑋‘𝑖) ∈ (Base‘𝐾))
12		lindfpropd.6	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
13	12	oveqrspc2v 7436	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝐾)) ∧ (𝑋‘𝑖) ∈ (Base‘𝐾))) → (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) = (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)))
14	6, 8, 11, 13	syl12anc 836	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) = (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)))
15		eqidd 2734	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
16		lindfpropd.1	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
17		ssidd 4006	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) ⊆ (Base‘𝐾))
18		lindfpropd.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
19		lindfpropd.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
20		eqidd 2734	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾)))
21		lindfpropd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
22		lindfpropd.l	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
23	15, 16, 17, 18, 19, 12, 20, 1, 21, 22	lsppropd 20629	. . . . . . . . . 10 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
24	23	fveq1d 6894	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
25	24	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
26	14, 25	eleq12d 2828	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → ((𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
27	26	notbid 318	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))})) → (¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
28	5, 27	raleqbidva 3328	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
29	28	ralbidva 3176	. . . 4 ⊢ ((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) → (∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
30	29	pm5.32da 580	. . 3 ⊢ (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
31	16	feq3d 6705	. . . 4 ⊢ (𝜑 → (𝑋:dom 𝑋⟶(Base‘𝐾) ↔ 𝑋:dom 𝑋⟶(Base‘𝐿)))
32	31	anbi1d 631	. . 3 ⊢ (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
33	30, 32	bitrd 279	. 2 ⊢ (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))) ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
34		lindfpropd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
35		eqid 2733	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
36		eqid 2733	. . . 4 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
37		eqid 2733	. . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
38		eqid 2733	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
39		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
40		eqid 2733	. . . 4 ⊢ (0_g‘(Scalar‘𝐾)) = (0_g‘(Scalar‘𝐾))
41	35, 36, 37, 38, 39, 40	islindf 21367	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
42	21, 34, 41	syl2anc 585	. 2 ⊢ (𝜑 → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0_g‘(Scalar‘𝐾))}) ¬ (𝑘( ·_𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
43		eqid 2733	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
44		eqid 2733	. . . 4 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
45		eqid 2733	. . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
46		eqid 2733	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
47		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
48		eqid 2733	. . . 4 ⊢ (0_g‘(Scalar‘𝐿)) = (0_g‘(Scalar‘𝐿))
49	43, 44, 45, 46, 47, 48	islindf 21367	. . 3 ⊢ ((𝐿 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
50	22, 34, 49	syl2anc 585	. 2 ⊢ (𝜑 → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0_g‘(Scalar‘𝐿))}) ¬ (𝑘( ·_𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
51	33, 42, 50	3bitr4d 311	1 ⊢ (𝜑 → (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∖ 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 +_gcplusg 17197 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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 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-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-lss 20543 df-lsp 20583 df-lindf 21361

This theorem is referenced by: lindspropd 32499

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