Theorem lindfpropd 33402

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 → (0g‘(Scalar‘𝐾)) = (0g‘(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 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
3	2	sneqd 4618	. . . . . . . 8 ⊢ (𝜑 → {(0g‘(Scalar‘𝐾))} = {(0g‘(Scalar‘𝐿))})
4	1, 3	difeq12d 4107	. . . . . . 7 ⊢ (𝜑 → ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}))
5	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) = ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}))
6		simplll 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝜑)
7		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}))
8	7	eldifad 3943	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → 𝑘 ∈ (Base‘(Scalar‘𝐾)))
9		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) → 𝑋:dom 𝑋⟶(Base‘𝐾))
10	9	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (𝑋‘𝑖) ∈ (Base‘𝐾))
11	10	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(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 7437	. . . . . . . . 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)) ∖ {(0g‘(Scalar‘𝐾))})) → (𝑘( ·𝑠 ‘𝐾)(𝑋‘𝑖)) = (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)))
15		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
16		lindfpropd.1	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
17		ssidd 3987	. . . . . . . . . . 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 2737	. . . . . . . . . . 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 20981	. . . . . . . . . 10 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
24	23	fveq1d 6883	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
25	24	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))})) → ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) = ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))
26	14, 25	eleq12d 2829	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(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)) ∖ {(0g‘(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 3315	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) ∧ 𝑖 ∈ dom 𝑋) → (∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
29	28	ralbidva 3162	. . . 4 ⊢ ((𝜑 ∧ 𝑋:dom 𝑋⟶(Base‘𝐾)) → (∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))) ↔ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖})))))
30	29	pm5.32da 579	. . 3 ⊢ (𝜑 → ((𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(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)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
31	16	feq3d 6698	. . . 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)) ∖ {(0g‘(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)) ∖ {(0g‘(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)) ∖ {(0g‘(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)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
34		lindfpropd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
35		eqid 2736	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
36		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
37		eqid 2736	. . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
38		eqid 2736	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
39		eqid 2736	. . . 4 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
40		eqid 2736	. . . 4 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
41	35, 36, 37, 38, 39, 40	islindf 21777	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
42	21, 34, 41	syl2anc 584	. 2 ⊢ (𝜑 → (𝑋 LIndF 𝐾 ↔ (𝑋:dom 𝑋⟶(Base‘𝐾) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐾)) ∖ {(0g‘(Scalar‘𝐾))}) ¬ (𝑘( ·𝑠 ‘𝐾)(𝑋‘𝑖)) ∈ ((LSpan‘𝐾)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
43		eqid 2736	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
44		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
45		eqid 2736	. . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
46		eqid 2736	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
47		eqid 2736	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
48		eqid 2736	. . . 4 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
49	43, 44, 45, 46, 47, 48	islindf 21777	. . 3 ⊢ ((𝐿 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(Scalar‘𝐿))}) ¬ (𝑘( ·𝑠 ‘𝐿)(𝑋‘𝑖)) ∈ ((LSpan‘𝐿)‘(𝑋 “ (dom 𝑋 ∖ {𝑖}))))))
50	22, 34, 49	syl2anc 584	. 2 ⊢ (𝜑 → (𝑋 LIndF 𝐿 ↔ (𝑋:dom 𝑋⟶(Base‘𝐿) ∧ ∀𝑖 ∈ dom 𝑋∀𝑘 ∈ ((Base‘(Scalar‘𝐿)) ∖ {(0g‘(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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ∖ cdif 3928  {csn 4606   class class class wbr 5124  dom cdm 5659   “ cima 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  Scalarcsca 17279   ·_𝑠 cvsca 17280  0_gc0g 17458  LSpanclspn 20933   LIndF clindf 21769

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-lss 20894  df-lsp 20934  df-lindf 21771

This theorem is referenced by:  lindspropd  33403