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Theorem lsppropd 18781
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 2641 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
43pweqd 4108 . . 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 18780 . . . . 5 (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
12 rabeq 3161 . . . . 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 4405 . . 3 (𝜑 {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠𝑡})
154, 14mpteq12dv 4653 . 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 2605 . . . 4 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2605 . . . 4 (LSubSp‘𝐾) = (LSubSp‘𝐾)
19 eqid 2605 . . . 4 (LSpan‘𝐾) = (LSpan‘𝐾)
2017, 18, 19lspfval 18736 . . 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 2605 . . . 4 (Base‘𝐿) = (Base‘𝐿)
24 eqid 2605 . . . 4 (LSubSp‘𝐿) = (LSubSp‘𝐿)
25 eqid 2605 . . . 4 (LSpan‘𝐿) = (LSpan‘𝐿)
2623, 24, 25lspfval 18736 . . 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 2649 1 (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535  𝒫 cpw 4103   cint 4400  cmpt 4633  cfv 5786  (class class class)co 6523  Basecbs 15637  +gcplusg 15710  Scalarcsca 15713   ·𝑠 cvsca 15714  LSubSpclss 18695  LSpanclspn 18734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-lss 18696  df-lsp 18735
This theorem is referenced by:  lbspropd  18862  lidlrsppropd  18993
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