MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsppropd Structured version   Visualization version   GIF version

Theorem lsppropd 19413
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 2815 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
43pweqd 4383 . . 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 19412 . . . . 5 (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
12 rabeq 3388 . . . . 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 4715 . . 3 (𝜑 {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
154, 14mpteq12dv 4969 . 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 2777 . . . 4 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2777 . . . 4 (LSubSp‘𝐾) = (LSubSp‘𝐾)
19 eqid 2777 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
2017, 18, 19lspfval 19368 . . 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 2777 . . . 4 (Base‘𝐿) = (Base‘𝐿)
24 eqid 2777 . . . 4 (LSubSp‘𝐿) = (LSubSp‘𝐿)
25 eqid 2777 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
2623, 24, 25lspfval 19368 . . 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 2823 1 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  {crab 3093  Vcvv 3397  wss 3791  𝒫 cpw 4378   cint 4710  cmpt 4965  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  Scalarcsca 16341   ·𝑠 cvsca 16342  LSubSpclss 19324  LSpanclspn 19366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-lss 19325  df-lsp 19367
This theorem is referenced by:  lbspropd  19494  lidlrsppropd  19627
  Copyright terms: Public domain W3C validator