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Theorem lbspropd 19018
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.)
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 (𝜑𝐾 ∈ V)
lbspropd.v2 (𝜑𝐿 ∈ V)
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

Proof of Theorem lbspropd
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplll 797 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝜑)
2 eldifi 3710 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) → 𝑣𝑃)
32adantl 482 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝑣𝑃)
4 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐵) → 𝑧𝐵)
54sselda 3583 . . . . . . . . . . . . . 14 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑢𝐵)
65adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → 𝑢𝐵)
7 lbspropd.s2 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
87oveqrspc2v 6627 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣𝑃𝑢𝐵)) → (𝑣( ·𝑠𝐾)𝑢) = (𝑣( ·𝑠𝐿)𝑢))
91, 3, 6, 8syl12anc 1321 . . . . . . . . . . . 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 6151 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘(Scalar‘𝐾))
1815, 17syl6eq 2671 . . . . . . . . . . . . . . 15 (𝜑𝑃 = (Base‘(Scalar‘𝐾)))
19 lbspropd.p2 . . . . . . . . . . . . . . . 16 (𝜑𝑃 = (Base‘𝐺))
20 lbspropd.g . . . . . . . . . . . . . . . . 17 𝐺 = (Scalar‘𝐿)
2120fveq2i 6151 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘(Scalar‘𝐿))
2219, 21syl6eq 2671 . . . . . . . . . . . . . . 15 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
23 lbspropd.v1 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ V)
24 lbspropd.v2 . . . . . . . . . . . . . . 15 (𝜑𝐿 ∈ V)
2510, 11, 12, 13, 14, 7, 18, 22, 23, 24lsppropd 18937 . . . . . . . . . . . . . 14 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
261, 25syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → (LSpan‘𝐾) = (LSpan‘𝐿))
2726fveq1d 6150 . . . . . . . . . . . 12 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) = ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢})))
289, 27eleq12d 2692 . . . . . . . . . . 11 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → ((𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
2928notbid 308 . . . . . . . . . 10 ((((𝜑𝑧𝐵) ∧ 𝑢𝑧) ∧ 𝑣 ∈ (𝑃 ∖ {(0g𝐹)})) → (¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
3029ralbidva 2979 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐾)𝑢) ∈ ((LSpan‘𝐾)‘(𝑧 ∖ {𝑢})) ↔ ∀𝑣 ∈ (𝑃 ∖ {(0g𝐹)}) ¬ (𝑣( ·𝑠𝐿)𝑢) ∈ ((LSpan‘𝐿)‘(𝑧 ∖ {𝑢}))))
3115ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑃 = (Base‘𝐹))
3231difeq1d 3705 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (𝑃 ∖ {(0g𝐹)}) = ((Base‘𝐹) ∖ {(0g𝐹)}))
3332raleqdv 3133 . . . . . . . . 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 761 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → 𝑃 = (Base‘𝐺))
35 lbspropd.a . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))
3615, 19, 35grpidpropd 17182 . . . . . . . . . . . . 13 (𝜑 → (0g𝐹) = (0g𝐺))
3736ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (0g𝐹) = (0g𝐺))
3837sneqd 4160 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → {(0g𝐹)} = {(0g𝐺)})
3934, 38difeq12d 3707 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑢𝑧) → (𝑃 ∖ {(0g𝐹)}) = ((Base‘𝐺) ∖ {(0g𝐺)}))
4039raleqdv 3133 . . . . . . . . 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 298 . . . . . . . 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 2979 . . . . . . 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 739 . . . . . 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 672 . . . . 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 3612 . . . . . 6 (𝜑 → (𝑧𝐵𝑧 ⊆ (Base‘𝐾)))
4645anbi1d 740 . . . . 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 3612 . . . . . 6 (𝜑 → (𝑧𝐵𝑧 ⊆ (Base‘𝐿)))
4825fveq1d 6150 . . . . . . . 8 (𝜑 → ((LSpan‘𝐾)‘𝑧) = ((LSpan‘𝐿)‘𝑧))
4910, 11eqtr3d 2657 . . . . . . . 8 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
5048, 49eqeq12d 2636 . . . . . . 7 (𝜑 → (((LSpan‘𝐾)‘𝑧) = (Base‘𝐾) ↔ ((LSpan‘𝐿)‘𝑧) = (Base‘𝐿)))
5150anbi1d 740 . . . . . 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 746 . . . . 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 298 . . . 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 1040 . . . 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 1040 . . . 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 303 . . 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 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
58 eqid 2621 . . . . 5 ( ·𝑠𝐾) = ( ·𝑠𝐾)
59 eqid 2621 . . . . 5 (Base‘𝐹) = (Base‘𝐹)
60 eqid 2621 . . . . 5 (LBasis‘𝐾) = (LBasis‘𝐾)
61 eqid 2621 . . . . 5 (LSpan‘𝐾) = (LSpan‘𝐾)
62 eqid 2621 . . . . 5 (0g𝐹) = (0g𝐹)
6357, 16, 58, 59, 60, 61, 62islbs 18995 . . . 4 (𝐾 ∈ V → (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 2621 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
66 eqid 2621 . . . . 5 ( ·𝑠𝐿) = ( ·𝑠𝐿)
67 eqid 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
68 eqid 2621 . . . . 5 (LBasis‘𝐿) = (LBasis‘𝐿)
69 eqid 2621 . . . . 5 (LSpan‘𝐿) = (LSpan‘𝐿)
70 eqid 2621 . . . . 5 (0g𝐺) = (0g𝐺)
7165, 20, 66, 67, 68, 69, 70islbs 18995 . . . 4 (𝐿 ∈ V → (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 300 . 2 (𝜑 → (𝑧 ∈ (LBasis‘𝐾) ↔ 𝑧 ∈ (LBasis‘𝐿)))
7473eqrdv 2619 1 (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  {csn 4148  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  LSpanclspn 18890  LBasisclbs 18993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-0g 16023  df-lss 18852  df-lsp 18891  df-lbs 18994
This theorem is referenced by: (None)
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