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Theorem lpolpolsatN 40664
Description: Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolpolsat.a 𝐴 = (LSAtomsβ€˜π‘Š)
lpolpolsat.p 𝑃 = (LPolβ€˜π‘Š)
lpolpolsat.w (πœ‘ β†’ π‘Š ∈ 𝑋)
lpolpolsat.o (πœ‘ β†’ βŠ₯ ∈ 𝑃)
lpolpolsat.q (πœ‘ β†’ 𝑄 ∈ 𝐴)
Assertion
Ref Expression
lpolpolsatN (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)

Proof of Theorem lpolpolsatN
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolpolsat.o . . 3 (πœ‘ β†’ βŠ₯ ∈ 𝑃)
2 lpolpolsat.w . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑋)
3 eqid 2731 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
4 eqid 2731 . . . . 5 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
5 eqid 2731 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
6 lpolpolsat.a . . . . 5 𝐴 = (LSAtomsβ€˜π‘Š)
7 eqid 2731 . . . . 5 (LSHypβ€˜π‘Š) = (LSHypβ€˜π‘Š)
8 lpolpolsat.p . . . . 5 𝑃 = (LPolβ€˜π‘Š)
93, 4, 5, 6, 7, 8islpolN 40658 . . . 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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( 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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( 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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
12 simpr3 1195 . . 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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))
13 lpolpolsat.q . . . 4 (πœ‘ β†’ 𝑄 ∈ 𝐴)
14 fveq2 6892 . . . . . . 7 (π‘₯ = 𝑄 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘„))
1514eleq1d 2817 . . . . . 6 (π‘₯ = 𝑄 β†’ (( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ↔ ( βŠ₯ β€˜π‘„) ∈ (LSHypβ€˜π‘Š)))
16 2fveq3 6897 . . . . . . 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 2747 . . . . . 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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) ↔ (( βŠ₯ β€˜π‘„) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
2019rspcv 3609 . . . 4 (𝑄 ∈ 𝐴 β†’ (βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) β†’ (( βŠ₯ β€˜π‘„) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
2113, 20syl 17 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 (( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯) β†’ (( βŠ₯ β€˜π‘„) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)))
22 simpr 484 . . 3 ((( βŠ₯ β€˜π‘„) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)
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‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄))
2411, 23mpd 15 1 (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘„)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086  βˆ€wal 1538   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βŸΆwf 6540  β€˜cfv 6544  Basecbs 17149  0gc0g 17390  LSubSpclss 20687  LSAtomsclsa 38148  LSHypclsh 38149  LPolclpoN 40655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8825  df-lpolN 40656
This theorem is referenced by: (None)
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