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Theorem lsppropd 20917
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 2770 . . . 4 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
43pweqd 4623 . . 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 20916 . . . . 5 (πœ‘ β†’ (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΏ))
1211rabeqdv 3446 . . . 4 (πœ‘ β†’ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑})
1312inteqd 4958 . . 3 (πœ‘ β†’ ∩ {𝑑 ∈ (LSubSpβ€˜πΎ) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ (LSubSpβ€˜πΏ) ∣ 𝑠 βŠ† 𝑑})
144, 13mpteq12dv 5243 . 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 2728 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
17 eqid 2728 . . . 4 (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΎ)
18 eqid 2728 . . . 4 (LSpanβ€˜πΎ) = (LSpanβ€˜πΎ)
1916, 17, 18lspfval 20871 . . 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 2728 . . . 4 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
23 eqid 2728 . . . 4 (LSubSpβ€˜πΏ) = (LSubSpβ€˜πΏ)
24 eqid 2728 . . . 4 (LSpanβ€˜πΏ) = (LSpanβ€˜πΏ)
2522, 23, 24lspfval 20871 . . 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 2778 1 (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430   βŠ† wss 3949  π’« cpw 4606  βˆ© cint 4953   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  +gcplusg 17242  Scalarcsca 17245   ·𝑠 cvsca 17246  LSubSpclss 20829  LSpanclspn 20869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-lss 20830  df-lsp 20870
This theorem is referenced by:  lbspropd  20998  lidlrsppropd  21153  lindfpropd  33130
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