Theorem lpolconN 42321

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	⊢ (𝜑 → 𝑊 ∈ 𝑋)
lpolcon.o	⊢ (𝜑 → ⊥ ∈ 𝑃)
lpolcon.x	⊢ (𝜑 → 𝑋 ⊆ 𝑉)
lpolcon.y	⊢ (𝜑 → 𝑌 ⊆ 𝑉)
lpolcon.c	⊢ (𝜑 → 𝑋 ⊆ 𝑌)

Assertion

Ref	Expression
lpolconN	⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))

Proof of Theorem lpolconN

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpolcon.o	. . 3 ⊢ (𝜑 → ⊥ ∈ 𝑃)
2		lpolcon.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
3		lpolcon.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
4		eqid 2765	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2765	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		eqid 2765	. . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7		eqid 2765	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolcon.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 42317	. . . 4 ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
10	2, 9	syl 18	. . 3 ⊢ (𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
11	1, 10	mpbid 235	. 2 ⊢ (𝜑 → ( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))
12		simpr2 1214	. . 3 ⊢ (( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)))
13		lpolcon.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
14		lpolcon.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑉)
15		lpolcon.c	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
16	13, 14, 15	3jca 1146	. . . 4 ⊢ (𝜑 → (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌))
17	3	fvexi 6899	. . . . . . 7 ⊢ 𝑉 ∈ V
18	17	elpw2 5307	. . . . . 6 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
19	13, 18	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑉)
20	17	elpw2 5307	. . . . . 6 ⊢ (𝑌 ∈ 𝒫 𝑉 ↔ 𝑌 ⊆ 𝑉)
21	14, 20	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝑉)
22		sseq1 3963	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑉 ↔ 𝑋 ⊆ 𝑉))
23		biidd 265	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 ⊆ 𝑉 ↔ 𝑦 ⊆ 𝑉))
24		sseq1 3963	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
25	22, 23, 24	3anbi123d 1464	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦)))
26		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
27	26	sseq2d 3970	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥) ↔ ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋)))
28	25, 27	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋))))
29		biidd 265	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 ⊆ 𝑉 ↔ 𝑋 ⊆ 𝑉))
30		sseq1 3963	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 ⊆ 𝑉 ↔ 𝑌 ⊆ 𝑉))
31		sseq2 3964	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑌))
32	29, 30, 31	3anbi123d 1464	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) ↔ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌)))
33		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
34	33	sseq1d 3969	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
35	32, 34	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
36	28, 35	sylan9bb 519	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
37	36	spc2gv 3561	. . . . 5 ⊢ ((𝑋 ∈ 𝒫 𝑉 ∧ 𝑌 ∈ 𝒫 𝑉) → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) → ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
38	19, 21, 37	syl2anc 596	. . . 4 ⊢ (𝜑 → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) → ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
39	16, 38	mpid 45	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
40	12, 39	syl5 35	. 2 ⊢ (𝜑 → (( ⊥ :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( ⊥ ‘𝑉) = {(0g‘𝑊)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( ⊥ ‘𝑥) ∈ (LSHyp‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
41	11, 40	mpd 16	1 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  𝒫 cpw 4564  {csn 4591  ⟶wf 6536  ‘cfv 6540  Basecbs 17293  0_gc0g 17516  LSubSpclss 21104  LSAtomsclsa 39808  LSHypclsh 39809  LPolclpoN 42314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-lpolN 42315

This theorem is used by: (None)