Theorem lindspropd 33443

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 3954	. . . 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 3433	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ V)
12	11	resiexd 7171	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝑧) ∈ V)
13	1, 3, 4, 5, 6, 7, 8, 9, 12	lindfpropd 33442	. . . 4 ⊢ (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿))
14	2, 13	anbi12d 633	. . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿)))
15		eqid 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	15	islinds 21789	. . . 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 2736	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
19	18	islinds 21789	. . . 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 2734	1 ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085   I cid 5525   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   LIndF clindf 21784  LIndSclinds 21785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-lss 20927  df-lsp 20967  df-lindf 21786  df-linds 21787

This theorem is referenced by:  fedgmullem2  33774