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Theorem lpolconN 37297
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 2771 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 eqid 2771 . . . . 5 (0g𝑊) = (0g𝑊)
6 eqid 2771 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2771 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolcon.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 37293 . . . 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 1235 . . 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 1122 . . . 4 (𝜑 → (𝑋𝑉𝑌𝑉𝑋𝑌))
17 fvex 6342 . . . . . . . 8 (Base‘𝑊) ∈ V
183, 17eqeltri 2846 . . . . . . 7 𝑉 ∈ V
1918elpw2 4959 . . . . . 6 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
2013, 19sylibr 224 . . . . 5 (𝜑𝑋 ∈ 𝒫 𝑉)
2118elpw2 4959 . . . . . 6 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
2214, 21sylibr 224 . . . . 5 (𝜑𝑌 ∈ 𝒫 𝑉)
23 sseq1 3775 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑉𝑋𝑉))
24 biidd 252 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑉𝑦𝑉))
25 sseq1 3775 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
2623, 24, 253anbi123d 1547 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑥𝑉𝑦𝑉𝑥𝑦) ↔ (𝑋𝑉𝑦𝑉𝑋𝑦)))
27 fveq2 6332 . . . . . . . . 9 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
2827sseq2d 3782 . . . . . . . 8 (𝑥 = 𝑋 → (( 𝑦) ⊆ ( 𝑥) ↔ ( 𝑦) ⊆ ( 𝑋)))
2926, 28imbi12d 333 . . . . . . 7 (𝑥 = 𝑋 → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋))))
30 biidd 252 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑉𝑋𝑉))
31 sseq1 3775 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦𝑉𝑌𝑉))
32 sseq2 3776 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
3330, 31, 323anbi123d 1547 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑋𝑉𝑦𝑉𝑋𝑦) ↔ (𝑋𝑉𝑌𝑉𝑋𝑌)))
34 fveq2 6332 . . . . . . . . 9 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
3534sseq1d 3781 . . . . . . . 8 (𝑦 = 𝑌 → (( 𝑦) ⊆ ( 𝑋) ↔ ( 𝑌) ⊆ ( 𝑋)))
3633, 35imbi12d 333 . . . . . . 7 (𝑦 = 𝑌 → (((𝑋𝑉𝑦𝑉𝑋𝑦) → ( 𝑦) ⊆ ( 𝑋)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3729, 36sylan9bb 499 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ↔ ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3837spc2gv 3447 . . . . 5 ((𝑋 ∈ 𝒫 𝑉𝑌 ∈ 𝒫 𝑉) → (∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) → ((𝑋𝑉𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))))
3920, 22, 38syl2anc 573 . . . 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 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  𝒫 cpw 4297  {csn 4316  wf 6027  cfv 6031  Basecbs 16064  0gc0g 16308  LSubSpclss 19142  LSAtomsclsa 34783  LSHypclsh 34784  LPolclpoN 37290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-lpolN 37291
This theorem is referenced by: (None)
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