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Theorem lsppropd 20494
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 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
lsspropd.b2 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
lsspropd.w (πœ‘ β†’ 𝐡 βŠ† π‘Š)
lsspropd.p ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
lsspropd.s1 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)
lsspropd.s2 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
lsspropd.p1 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))
lsspropd.p2 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))
lsppropd.v1 (πœ‘ β†’ 𝐾 ∈ 𝑋)
lsppropd.v2 (πœ‘ β†’ 𝐿 ∈ π‘Œ)
Assertion
Ref Expression
lsppropd (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦   πœ‘,π‘₯,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝑃,𝑦
Allowed substitution hints:   𝑋(π‘₯,𝑦)   π‘Œ(π‘₯,𝑦)

Proof of Theorem lsppropd
Dummy variables 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsspropd.b1 . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
2 lsspropd.b2 . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
31, 2eqtr3d 2775 . . . 4 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
43pweqd 4578 . . 3 (πœ‘ β†’ 𝒫 (Baseβ€˜πΎ) = 𝒫 (Baseβ€˜πΏ))
5 lsspropd.w . . . . . 6 (πœ‘ β†’ 𝐡 βŠ† π‘Š)
6 lsspropd.p . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
7 lsspropd.s1 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)
8 lsspropd.s2 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
9 lsspropd.p1 . . . . . 6 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))
10 lsspropd.p2 . . . . . 6 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))
111, 2, 5, 6, 7, 8, 9, 10lsspropd 20493 . . . . 5 (πœ‘ β†’ (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΏ))
1211rabeqdv 3421 . . . 4 (πœ‘ β†’ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑})
1312inteqd 4913 . . 3 (πœ‘ β†’ ∩ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑})
144, 13mpteq12dv 5197 . 2 (πœ‘ β†’ (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 (Baseβ€˜πΏ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑}))
15 lsppropd.v1 . . 3 (πœ‘ β†’ 𝐾 ∈ 𝑋)
16 eqid 2733 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
17 eqid 2733 . . . 4 (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΎ)
18 eqid 2733 . . . 4 (LSpanβ€˜πΎ) = (LSpanβ€˜πΎ)
1916, 17, 18lspfval 20449 . . 3 (𝐾 ∈ 𝑋 β†’ (LSpanβ€˜πΎ) = (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑}))
2015, 19syl 17 . 2 (πœ‘ β†’ (LSpanβ€˜πΎ) = (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑}))
21 lsppropd.v2 . . 3 (πœ‘ β†’ 𝐿 ∈ π‘Œ)
22 eqid 2733 . . . 4 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
23 eqid 2733 . . . 4 (LSubSpβ€˜πΏ) = (LSubSpβ€˜πΏ)
24 eqid 2733 . . . 4 (LSpanβ€˜πΏ) = (LSpanβ€˜πΏ)
2522, 23, 24lspfval 20449 . . 3 (𝐿 ∈ π‘Œ β†’ (LSpanβ€˜πΏ) = (𝑠 ∈ 𝒫 (Baseβ€˜πΏ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑}))
2621, 25syl 17 . 2 (πœ‘ β†’ (LSpanβ€˜πΏ) = (𝑠 ∈ 𝒫 (Baseβ€˜πΏ) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑}))
2714, 20, 263eqtr4d 2783 1 (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3406   βŠ† wss 3911  π’« cpw 4561  βˆ© cint 4908   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  LSubSpclss 20407  LSpanclspn 20447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-lss 20408  df-lsp 20448
This theorem is referenced by:  lbspropd  20575  lidlrsppropd  20716  lindfpropd  32217
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