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Theorem lsppropd 18999
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 2656 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
43pweqd 4154 . . 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 18998 . . . . 5 (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
12 rabeq 3187 . . . . 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 4471 . . 3 (𝜑 {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
154, 14mpteq12dv 4724 . 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 2620 . . . 4 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2620 . . . 4 (LSubSp‘𝐾) = (LSubSp‘𝐾)
19 eqid 2620 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
2017, 18, 19lspfval 18954 . . 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 2620 . . . 4 (Base‘𝐿) = (Base‘𝐿)
24 eqid 2620 . . . 4 (LSubSp‘𝐿) = (LSubSp‘𝐿)
25 eqid 2620 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
2623, 24, 25lspfval 18954 . . 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 2664 1 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  wss 3567  𝒫 cpw 4149   cint 4466  cmpt 4720  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Scalarcsca 15925   ·𝑠 cvsca 15926  LSubSpclss 18913  LSpanclspn 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-lss 18914  df-lsp 18953
This theorem is referenced by:  lbspropd  19080  lidlrsppropd  19211
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