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Theorem lsppropd 21017
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.) (Revised by AV, 24-Apr-2024.)
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)))
lsppropd.v1 (𝜑𝐾𝑋)
lsppropd.v2 (𝜑𝐿𝑌)
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
Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,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 2779 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
43pweqd 4617 . . 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 21016 . . . . 5 (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
1211rabeqdv 3452 . . . 4 (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
1312inteqd 4951 . . 3 (𝜑 {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
144, 13mpteq12dv 5233 . 2 (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
15 lsppropd.v1 . . 3 (𝜑𝐾𝑋)
16 eqid 2737 . . . 4 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2737 . . . 4 (LSubSp‘𝐾) = (LSubSp‘𝐾)
18 eqid 2737 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
1916, 17, 18lspfval 20971 . . 3 (𝐾𝑋 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}))
2015, 19syl 17 . 2 (𝜑 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡}))
21 lsppropd.v2 . . 3 (𝜑𝐿𝑌)
22 eqid 2737 . . . 4 (Base‘𝐿) = (Base‘𝐿)
23 eqid 2737 . . . 4 (LSubSp‘𝐿) = (LSubSp‘𝐿)
24 eqid 2737 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
2522, 23, 24lspfval 20971 . . 3 (𝐿𝑌 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
2621, 25syl 17 . 2 (𝜑 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡}))
2714, 20, 263eqtr4d 2787 1 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600   cint 4946  cmpt 5225  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  LSubSpclss 20929  LSpanclspn 20969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-lss 20930  df-lsp 20970
This theorem is referenced by:  lbspropd  21098  lidlrsppropd  21254  lindfpropd  33410
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