Theorem lindspropd 31479

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 3949	. . . 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 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ V)
12	11	resiexd 7074	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝑧) ∈ V)
13	1, 3, 4, 5, 6, 7, 8, 9, 12	lindfpropd 31478	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿))
14	2, 13	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
15		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	15	islinds 20926	. . . 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 2738	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
19	18	islinds 20926	. . . 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 310	. 2 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ 𝑧 ∈ (LIndS‘𝐿)))
22	21	eqrdv 2736	1 ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070   I cid 5479   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067   LIndF clindf 20921  LIndSclinds 20922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-lss 20109  df-lsp 20149  df-lindf 20923  df-linds 20924

This theorem is referenced by:  fedgmullem2  31613