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Theorem lpolconN 37508
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 2799 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2799 . . . . 5 (0g𝑊) = (0g𝑊)
6 eqid 2799 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2799 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolcon.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 37504 . . . 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 224 . 2 (𝜑 → ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr2 1251 . . 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 1159 . . . 4 (𝜑 → (𝑋𝑉𝑌𝑉𝑋𝑌))
173fvexi 6425 . . . . . . 7 𝑉 ∈ V
1817elpw2 5020 . . . . . 6 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1913, 18sylibr 226 . . . . 5 (𝜑𝑋 ∈ 𝒫 𝑉)
2017elpw2 5020 . . . . . 6 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
2114, 20sylibr 226 . . . . 5 (𝜑𝑌 ∈ 𝒫 𝑉)
22 sseq1 3822 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑉𝑋𝑉))
23 biidd 254 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑉𝑦𝑉))
24 sseq1 3822 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
2522, 23, 243anbi123d 1561 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥𝑉𝑦𝑉𝑥𝑦) ↔ (𝑋𝑉𝑦𝑉𝑋𝑦)))
26 fveq2 6411 . . . . . . . . 9 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
2726sseq2d 3829 . . . . . . . 8 (𝑥 = 𝑋 → (( 𝑦) ⊆ ( 𝑥) ↔ ( 𝑦) ⊆ ( 𝑋)))
2825, 27imbi12d 336 . . . . . . 7 (𝑥 = 𝑋 → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋))))
29 biidd 254 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑉𝑋𝑉))
30 sseq1 3822 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦𝑉𝑌𝑉))
31 sseq2 3823 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
3229, 30, 313anbi123d 1561 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋𝑉𝑦𝑉𝑋𝑦) ↔ (𝑋𝑉𝑌𝑉𝑋𝑌)))
33 fveq2 6411 . . . . . . . . 9 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
3433sseq1d 3828 . . . . . . . 8 (𝑦 = 𝑌 → (( 𝑦) ⊆ ( 𝑋) ↔ ( 𝑌) ⊆ ( 𝑋)))
3532, 34imbi12d 336 . . . . . . 7 (𝑦 = 𝑌 → (((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3628, 35sylan9bb 506 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3736spc2gv 3484 . . . . 5 ((𝑋 ∈ 𝒫 𝑉𝑌 ∈ 𝒫 𝑉) → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3819, 21, 37syl2anc 580 . . . 4 (𝜑 → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3916, 38mpid 44 . . 3 (𝜑 → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ( 𝑌) ⊆ ( 𝑋)))
4012, 39syl5 34 . 2 (𝜑 → (( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ( 𝑌) ⊆ ( 𝑋)))
4111, 40mpd 15 1 (𝜑 → ( 𝑌) ⊆ ( 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wcel 2157  wral 3089  wss 3769  𝒫 cpw 4349  {csn 4368  wf 6097  cfv 6101  Basecbs 16184  0gc0g 16415  LSubSpclss 19250  LSAtomsclsa 34995  LSHypclsh 34996  LPolclpoN 37501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-lpolN 37502
This theorem is referenced by: (None)
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