Theorem lindspropd 33330

Description: Property deduction for linearly independent sets. (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)

Assertion

Ref	Expression
lindspropd	⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

Distinct variable groups:   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem lindspropd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lindfpropd.1	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
2	1	sseq2d 3970	. . . 4 ⊢ (𝜑 → (𝑧 ⊆ (Base‘𝐾) ↔ 𝑧 ⊆ (Base‘𝐿)))
3		lindfpropd.2	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))
4		lindfpropd.3	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
5		lindfpropd.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
6		lindfpropd.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
7		lindfpropd.6	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
8		lindfpropd.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
9		lindfpropd.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
10		vex 3442	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ V)
12	11	resiexd 7156	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝑧) ∈ V)
13	1, 3, 4, 5, 6, 7, 8, 9, 12	lindfpropd 33329	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿))
14	2, 13	anbi12d 632	. . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
15		eqid 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	15	islinds 21734	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾)))
17	8, 16	syl 17	. . 3 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾)))
18		eqid 2729	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
19	18	islinds 21734	. . . 4 ⊢ (𝐿 ∈ 𝑊 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
20	9, 19	syl 17	. . 3 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
21	14, 17, 20	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ 𝑧 ∈ (LIndS‘𝐿)))
22	21	eqrdv 2727	1 ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ⊆ wss 3905   class class class wbr 5095   I cid 5517   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361   LIndF clindf 21729  LIndSclinds 21730

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-lss 20853  df-lsp 20893  df-lindf 21731  df-linds 21732

This theorem is referenced by:  fedgmullem2  33602