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Theorem lsppropd 18839
 Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
lsspropd.b1 (𝜑𝐵 = (Base‘𝐾))
lsspropd.b2 (𝜑𝐵 = (Base‘𝐿))
lsspropd.w (𝜑𝐵𝑊)
lsspropd.p ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
lsspropd.s1 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)
lsspropd.s2 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
lsspropd.p1 (𝜑𝑃 = (Base‘(Scalar‘𝐾)))
lsspropd.p2 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
lsspropd.v1 (𝜑𝐾 ∈ V)
lsspropd.v2 (𝜑𝐿 ∈ V)
Assertion
Ref Expression
lsppropd (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦   𝑥,𝐿,𝑦   𝑥,𝑃,𝑦

Proof of Theorem lsppropd
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsspropd.b1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
2 lsspropd.b2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
31, 2eqtr3d 2646 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
43pweqd 4113 . . 3 (𝜑 → 𝒫 (Base‘𝐾) = 𝒫 (Base‘𝐿))
5 lsspropd.w . . . . . 6 (𝜑𝐵𝑊)
6 lsspropd.p . . . . . 6 ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
7 lsspropd.s1 . . . . . 6 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)
8 lsspropd.s2 . . . . . 6 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
9 lsspropd.p1 . . . . . 6 (𝜑𝑃 = (Base‘(Scalar‘𝐾)))
10 lsspropd.p2 . . . . . 6 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
111, 2, 5, 6, 7, 8, 9, 10lsspropd 18838 . . . . 5 (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
12 rabeq 3166 . . . . 5 ((LSubSp‘𝐾) = (LSubSp‘𝐿) → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
1311, 12syl 17 . . . 4 (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
1413inteqd 4415 . . 3 (𝜑 {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
154, 14mpteq12dv 4663 . 2 (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
16 lsspropd.v1 . . 3 (𝜑𝐾 ∈ V)
17 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2610 . . . 4 (LSubSp‘𝐾) = (LSubSp‘𝐾)
19 eqid 2610 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
2017, 18, 19lspfval 18794 . . 3 (𝐾 ∈ V → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}))
2116, 20syl 17 . 2 (𝜑 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}))
22 lsspropd.v2 . . 3 (𝜑𝐿 ∈ V)
23 eqid 2610 . . . 4 (Base‘𝐿) = (Base‘𝐿)
24 eqid 2610 . . . 4 (LSubSp‘𝐿) = (LSubSp‘𝐿)
25 eqid 2610 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
2623, 24, 25lspfval 18794 . . 3 (𝐿 ∈ V → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
2722, 26syl 17 . 2 (𝜑 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
2815, 21, 273eqtr4d 2654 1 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∩ cint 4410   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  LSubSpclss 18753  LSpanclspn 18792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-lss 18754  df-lsp 18793 This theorem is referenced by:  lbspropd  18920  lidlrsppropd  19051
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