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Theorem lbspropd 19862
 Description: If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
Hypotheses
Ref Expression
lbspropd.b1 (𝜑𝐵 = (Base‘𝐾))
lbspropd.b2 (𝜑𝐵 = (Base‘𝐿))
lbspropd.w (𝜑𝐵𝑊)
lbspropd.p ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
lbspropd.s1 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)
lbspropd.s2 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
lbspropd.f 𝐹 = (Scalar‘𝐾)
lbspropd.g 𝐺 = (Scalar‘𝐿)
lbspropd.p1 (𝜑𝑃 = (Base‘𝐹))
lbspropd.p2 (𝜑𝑃 = (Base‘𝐺))
lbspropd.a ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))
lbspropd.v1 (𝜑𝐾𝑋)
lbspropd.v2 (𝜑𝐿𝑌)
Assertion
Ref Expression
lbspropd (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝑃,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem lbspropd
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplll 774 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝜑)
2 eldifi 4078 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) → 𝑣𝑃)
32adantl 485 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝑣𝑃)
4 simpr 488 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐵) → 𝑧𝐵)
54sselda 3942 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑢𝐵)
65adantr 484 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝑢𝐵)
7 lbspropd.s2 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
87oveqrspc2v 7167 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣𝑃𝑢𝐵)) → (𝑣( ·𝑠𝐾)𝑢) = (𝑣( ·𝑠𝐿)𝑢))
91, 3, 6, 8syl12anc 835 . . . . . . . . . . . 12 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → (𝑣( ·𝑠𝐾)𝑢) = (𝑣( ·𝑠𝐿)𝑢))
10 lbspropd.b1 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (Base‘𝐾))
11 lbspropd.b2 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (Base‘𝐿))
12 lbspropd.w . . . . . . . . . . . . . . 15 (𝜑𝐵𝑊)
13 lbspropd.p . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
14 lbspropd.s1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)
15 lbspropd.p1 . . . . . . . . . . . . . . . 16 (𝜑𝑃 = (Base‘𝐹))
16 lbspropd.f . . . . . . . . . . . . . . . . 17 𝐹 = (Scalar‘𝐾)
1716fveq2i 6655 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘(Scalar‘𝐾))
1815, 17syl6eq 2873 . . . . . . . . . . . . . . 15 (𝜑𝑃 = (Base‘(Scalar‘𝐾)))
19 lbspropd.p2 . . . . . . . . . . . . . . . 16 (𝜑𝑃 = (Base‘𝐺))
20 lbspropd.g . . . . . . . . . . . . . . . . 17 𝐺 = (Scalar‘𝐿)
2120fveq2i 6655 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘(Scalar‘𝐿))
2219, 21syl6eq 2873 . . . . . . . . . . . . . . 15 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
23 lbspropd.v1 . . . . . . . . . . . . . . 15 (𝜑𝐾𝑋)
24 lbspropd.v2 . . . . . . . . . . . . . . 15 (𝜑𝐿𝑌)
2510, 11, 12, 13, 14, 7, 18, 22, 23, 24lsppropd 19781 . . . . . . . . . . . . . 14 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
261, 25syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → (LSpan‘𝐾) = (LSpan‘𝐿))
2726fveq1d 6654 . . . . . . . . . . . 12 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) = ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))
289, 27eleq12d 2908 . . . . . . . . . . 11 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → ((𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
2928notbid 321 . . . . . . . . . 10 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → (¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
3029ralbidva 3186 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
3115ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑃 = (Base‘𝐹))
3231difeq1d 4073 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (𝑃 ∖ {(0g𝐹)}) = ((Base‘𝐹) ∖ {(0g𝐹)}))
3332raleqdv 3392 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))))
3419ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑃 = (Base‘𝐺))
35 lbspropd.a . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))
3615, 19, 35grpidpropd 17863 . . . . . . . . . . . . 13 (𝜑 → (0g𝐹) = (0g𝐺))
3736ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (0g𝐹) = (0g𝐺))
3837sneqd 4551 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → {(0g𝐹)} = {(0g𝐺)})
3934, 38difeq12d 4075 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (𝑃 ∖ {(0g𝐹)}) = ((Base‘𝐺) ∖ {(0g𝐺)}))
4039raleqdv 3392 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
4130, 33, 403bitr3d 312 . . . . . . . 8 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (∀𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
4241ralbidva 3186 . . . . . . 7 ((𝜑𝑧𝐵) → (∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
4342anbi2d 631 . . . . . 6 ((𝜑𝑧𝐵) → ((((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
4443pm5.32da 582 . . . . 5 (𝜑 → ((𝑧𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
4510sseq2d 3974 . . . . . 6 (𝜑 → (𝑧𝐵𝑧 ⊆ (Base‘𝐾)))
4645anbi1d 632 . . . . 5 (𝜑 → ((𝑧𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))))))
4711sseq2d 3974 . . . . . 6 (𝜑 → (𝑧𝐵𝑧 ⊆ (Base‘𝐿)))
4825fveq1d 6654 . . . . . . . 8 (𝜑 → ((LSpan‘𝐾)‘𝑧) = ((LSpan‘𝐿)‘𝑧))
4910, 11eqtr3d 2859 . . . . . . . 8 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
5048, 49eqeq12d 2838 . . . . . . 7 (𝜑 → (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ↔ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿)))
5150anbi1d 632 . . . . . 6 (𝜑 → ((((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))) ↔ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
5247, 51anbi12d 633 . . . . 5 (𝜑 → ((𝑧𝐵 ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
5344, 46, 523bitr3d 312 . . . 4 (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))))
54 3anass 1092 . . . 4 ((𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
55 3anass 1092 . . . 4 ((𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ (((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
5653, 54, 553bitr4g 317 . . 3 (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢}))) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
57 eqid 2822 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
58 eqid 2822 . . . . 5 ( ·𝑠𝐾) = ( ·𝑠𝐾)
59 eqid 2822 . . . . 5 (Base‘𝐹) = (Base‘𝐹)
60 eqid 2822 . . . . 5 (LBasis‘𝐾) = (LBasis‘𝐾)
61 eqid 2822 . . . . 5 (LSpan‘𝐾) = (LSpan‘𝐾)
62 eqid 2822 . . . . 5 (0g𝐹) = (0g𝐹)
6357, 16, 58, 59, 60, 61, 62islbs 19839 . . . 4 (𝐾𝑋 → (𝑧 ∈ (LBasis‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
6423, 63syl 17 . . 3 (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐹) ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})))))
65 eqid 2822 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
66 eqid 2822 . . . . 5 ( ·𝑠𝐿) = ( ·𝑠𝐿)
67 eqid 2822 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
68 eqid 2822 . . . . 5 (LBasis‘𝐿) = (LBasis‘𝐿)
69 eqid 2822 . . . . 5 (LSpan‘𝐿) = (LSpan‘𝐿)
70 eqid 2822 . . . . 5 (0g𝐺) = (0g𝐺)
7165, 20, 66, 67, 68, 69, 70islbs 19839 . . . 4 (𝐿𝑌 → (𝑧 ∈ (LBasis‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
7224, 71syl 17 . . 3 (𝜑 → (𝑧 ∈ (LBasis‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿) ∧ ∀𝑢𝑧𝑣 ∈ ((Base‘𝐺) ∖ {(0g𝐺)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))))
7356, 64, 723bitr4d 314 . 2 (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ 𝑧 ∈ (LBasis‘𝐿)))
7473eqrdv 2820 1 (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130   ∖ cdif 3905   ⊆ wss 3908  {csn 4539  ‘cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Scalarcsca 16559   ·𝑠 cvsca 16560  0gc0g 16704  LSpanclspn 19734  LBasisclbs 19837 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-0g 16706  df-lss 19695  df-lsp 19735  df-lbs 19838 This theorem is referenced by:  dimpropd  31064
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