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Theorem lpolconN 36256
 Description: Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolcon.v 𝑉 = (Base‘𝑊)
lpolcon.p 𝑃 = (LPol‘𝑊)
lpolcon.w (𝜑𝑊𝑋)
lpolcon.o (𝜑𝑃)
lpolcon.x (𝜑𝑋𝑉)
lpolcon.y (𝜑𝑌𝑉)
lpolcon.c (𝜑𝑋𝑌)
Assertion
Ref Expression
lpolconN (𝜑 → ( 𝑌) ⊆ ( 𝑋))

Proof of Theorem lpolconN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolcon.o . . 3 (𝜑𝑃)
2 lpolcon.w . . . 4 (𝜑𝑊𝑋)
3 lpolcon.v . . . . 5 𝑉 = (Base‘𝑊)
4 eqid 2621 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2621 . . . . 5 (0g𝑊) = (0g𝑊)
6 eqid 2621 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2621 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolcon.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 36252 . . . 4 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
102, 9syl 17 . . 3 (𝜑 → ( 𝑃 ↔ ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
111, 10mpbid 222 . 2 (𝜑 → ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr2 1066 . . 3 (( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)))
13 lpolcon.x . . . . 5 (𝜑𝑋𝑉)
14 lpolcon.y . . . . 5 (𝜑𝑌𝑉)
15 lpolcon.c . . . . 5 (𝜑𝑋𝑌)
1613, 14, 153jca 1240 . . . 4 (𝜑 → (𝑋𝑉𝑌𝑉𝑋𝑌))
17 fvex 6158 . . . . . . . 8 (Base‘𝑊) ∈ V
183, 17eqeltri 2694 . . . . . . 7 𝑉 ∈ V
1918elpw2 4788 . . . . . 6 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
2013, 19sylibr 224 . . . . 5 (𝜑𝑋 ∈ 𝒫 𝑉)
2118elpw2 4788 . . . . . 6 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
2214, 21sylibr 224 . . . . 5 (𝜑𝑌 ∈ 𝒫 𝑉)
23 sseq1 3605 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑉𝑋𝑉))
24 biidd 252 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑉𝑦𝑉))
25 sseq1 3605 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
2623, 24, 253anbi123d 1396 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥𝑉𝑦𝑉𝑥𝑦) ↔ (𝑋𝑉𝑦𝑉𝑋𝑦)))
27 fveq2 6148 . . . . . . . . 9 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
2827sseq2d 3612 . . . . . . . 8 (𝑥 = 𝑋 → (( 𝑦) ⊆ ( 𝑥) ↔ ( 𝑦) ⊆ ( 𝑋)))
2926, 28imbi12d 334 . . . . . . 7 (𝑥 = 𝑋 → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋))))
30 biidd 252 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑉𝑋𝑉))
31 sseq1 3605 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦𝑉𝑌𝑉))
32 sseq2 3606 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
3330, 31, 323anbi123d 1396 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋𝑉𝑦𝑉𝑋𝑦) ↔ (𝑋𝑉𝑌𝑉𝑋𝑌)))
34 fveq2 6148 . . . . . . . . 9 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
3534sseq1d 3611 . . . . . . . 8 (𝑦 = 𝑌 → (( 𝑦) ⊆ ( 𝑋) ↔ ( 𝑌) ⊆ ( 𝑋)))
3633, 35imbi12d 334 . . . . . . 7 (𝑦 = 𝑌 → (((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3729, 36sylan9bb 735 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3837spc2gv 3282 . . . . 5 ((𝑋 ∈ 𝒫 𝑉𝑌 ∈ 𝒫 𝑉) → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3920, 22, 38syl2anc 692 . . . 4 (𝜑 → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
4016, 39mpid 44 . . 3 (𝜑 → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ( 𝑌) ⊆ ( 𝑋)))
4112, 40syl5 34 . 2 (𝜑 → (( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ( 𝑌) ⊆ ( 𝑋)))
4211, 41mpd 15 1 (𝜑 → ( 𝑌) ⊆ ( 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036  ∀wal 1478   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ⊆ wss 3555  𝒫 cpw 4130  {csn 4148  ⟶wf 5843  ‘cfv 5847  Basecbs 15781  0gc0g 16021  LSubSpclss 18851  LSAtomsclsa 33741  LSHypclsh 33742  LPolclpoN 36249 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-lpolN 36250 This theorem is referenced by: (None)
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