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Theorem lpolsatN 40872
Description: The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolsat.a 𝐴 = (LSAtomsβ€˜π‘Š)
lpolsat.h 𝐻 = (LSHypβ€˜π‘Š)
lpolsat.p 𝑃 = (LPolβ€˜π‘Š)
lpolsat.w (πœ‘ β†’ π‘Š ∈ 𝑋)
lpolsat.o (πœ‘ β†’ βŠ₯ ∈ 𝑃)
lpolsat.q (πœ‘ β†’ 𝑄 ∈ 𝐴)
Assertion
Ref Expression
lpolsatN (πœ‘ β†’ ( βŠ₯ β€˜π‘„) ∈ 𝐻)

Proof of Theorem lpolsatN
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolsat.o . . 3 (πœ‘ β†’ βŠ₯ ∈ 𝑃)
2 lpolsat.w . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑋)
3 eqid 2726 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
4 eqid 2726 . . . . 5 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
5 eqid 2726 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
6 lpolsat.a . . . . 5 𝐴 = (LSAtomsβ€˜π‘Š)
7 lpolsat.h . . . . 5 𝐻 = (LSHypβ€˜π‘Š)
8 lpolsat.p . . . . 5 𝑃 = (LPolβ€˜π‘Š)
93, 4, 5, 6, 7, 8islpolN 40867 . . . 4 (π‘Š ∈ 𝑋 β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 (Baseβ€˜π‘Š)⟢(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘Š) ∧ 𝑦 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
102, 9syl 17 . . 3 (πœ‘ β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 (Baseβ€˜π‘Š)⟢(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘Š) ∧ 𝑦 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
111, 10mpbid 231 . 2 (πœ‘ β†’ ( βŠ₯ :𝒫 (Baseβ€˜π‘Š)⟢(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘Š) ∧ 𝑦 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
12 simpr3 1193 . . 3 (( βŠ₯ :𝒫 (Baseβ€˜π‘Š)⟢(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘Š) ∧ 𝑦 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))
13 lpolsat.q . . . 4 (πœ‘ β†’ 𝑄 ∈ 𝐴)
14 fveq2 6885 . . . . . . 7 (π‘₯ = 𝑄 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘„))
1514eleq1d 2812 . . . . . 6 (π‘₯ = 𝑄 β†’ (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ↔ ( βŠ₯ β€˜π‘„) ∈ 𝐻))
16 2fveq3 6890 . . . . . . 7 (π‘₯ = 𝑄 β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)))
17 id 22 . . . . . . 7 (π‘₯ = 𝑄 β†’ π‘₯ = 𝑄)
1816, 17eqeq12d 2742 . . . . . 6 (π‘₯ = 𝑄 β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄))
1915, 18anbi12d 630 . . . . 5 (π‘₯ = 𝑄 β†’ ((( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) ↔ (( βŠ₯ β€˜π‘„) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
2019rspcv 3602 . . . 4 (𝑄 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) β†’ (( βŠ₯ β€˜π‘„) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
2113, 20syl 17 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) β†’ (( βŠ₯ β€˜π‘„) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
22 simpl 482 . . 3 ((( βŠ₯ β€˜π‘„) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄) β†’ ( βŠ₯ β€˜π‘„) ∈ 𝐻)
2312, 21, 22syl56 36 . 2 (πœ‘ β†’ (( βŠ₯ :𝒫 (Baseβ€˜π‘Š)⟢(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† (Baseβ€˜π‘Š) ∧ 𝑦 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ 𝐻 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ ( βŠ₯ β€˜π‘„) ∈ 𝐻))
2411, 23mpd 15 1 (πœ‘ β†’ ( βŠ₯ β€˜π‘„) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943  π’« cpw 4597  {csn 4623  βŸΆwf 6533  β€˜cfv 6537  Basecbs 17153  0gc0g 17394  LSubSpclss 20778  LSAtomsclsa 38357  LSHypclsh 38358  LPolclpoN 40864
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-lpolN 40865
This theorem is referenced by: (None)
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