Theorem lindfpropd 33339

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 4583	. . . . . . . 8 ⊢ (𝜑 → {(0g‘(Scalar‘𝐾))} = {(0g‘(Scalar‘𝐿))})
4	1, 3	difeq12d 4072	. . . . . . 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 3909	. . . . . . . . 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 7012	. . . . . . . . . 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 7368	. . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
16		lindfpropd.1	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
17		ssidd 3953	. . . . . . . . . . 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 2732	. . . . . . . . . . 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 20947	. . . . . . . . . 10 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
24	23	fveq1d 6819	. . . . . . . . 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 2825	. . . . . . 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 3298	. . . . 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 3153	. . . 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 6631	. . . 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 2731	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
36		eqid 2731	. . . 4 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
37		eqid 2731	. . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
38		eqid 2731	. . . 4 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
39		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
40		eqid 2731	. . . 4 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
41	35, 36, 37, 38, 39, 40	islindf 21744	. . 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 2731	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
44		eqid 2731	. . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
45		eqid 2731	. . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
46		eqid 2731	. . . 4 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
47		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
48		eqid 2731	. . . 4 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
49	43, 44, 45, 46, 47, 48	islindf 21744	. . 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 1541   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3894  {csn 4571   class class class wbr 5086  dom cdm 5611   “ cima 5614  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  +_gcplusg 17156  Scalarcsca 17159   ·_𝑠 cvsca 17160  0_gc0g 17338  LSpanclspn 20899   LIndF clindf 21736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-lss 20860  df-lsp 20900  df-lindf 21738

This theorem is referenced by:  lindspropd  33340