Theorem lpolconN 41490

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 2736	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
5		eqid 2736	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
6		eqid 2736	. . . . 5 ⊢ (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7		eqid 2736	. . . . 5 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
8		lpolcon.p	. . . . 5 ⊢ 𝑃 = (LPol‘𝑊)
9	3, 4, 5, 6, 7, 8	islpolN 41486	. . . 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 17	. . 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 232	. 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 1195	. . 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 1128	. . . 4 ⊢ (𝜑 → (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌))
17	3	fvexi 6919	. . . . . . 7 ⊢ 𝑉 ∈ V
18	17	elpw2 5333	. . . . . 6 ⊢ (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 ⊆ 𝑉)
19	13, 18	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑉)
20	17	elpw2 5333	. . . . . 6 ⊢ (𝑌 ∈ 𝒫 𝑉 ↔ 𝑌 ⊆ 𝑉)
21	14, 20	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝑉)
22		sseq1 4008	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑉 ↔ 𝑋 ⊆ 𝑉))
23		biidd 262	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 ⊆ 𝑉 ↔ 𝑦 ⊆ 𝑉))
24		sseq1 4008	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
25	22, 23, 24	3anbi123d 1437	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) ↔ (𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦)))
26		fveq2 6905	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
27	26	sseq2d 4015	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥) ↔ ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋)))
28	25, 27	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋))))
29		biidd 262	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 ⊆ 𝑉 ↔ 𝑋 ⊆ 𝑉))
30		sseq1 4008	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 ⊆ 𝑉 ↔ 𝑌 ⊆ 𝑉))
31		sseq2 4009	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑌))
32	29, 30, 31	3anbi123d 1437	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) ↔ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌)))
33		fveq2 6905	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
34	33	sseq1d 4014	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
35	32, 34	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝑋 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑋)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
36	28, 35	sylan9bb 509	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ↔ ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
37	36	spc2gv 3599	. . . . 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 584	. . . 4 ⊢ (𝜑 → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) → ((𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))))
39	16, 38	mpid 44	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
40	12, 39	syl5 34	. 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 15	1 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ⊆ wss 3950  𝒫 cpw 4599  {csn 4625  ⟶wf 6556  ‘cfv 6560  Basecbs 17248  0_gc0g 17485  LSubSpclss 20930  LSAtomsclsa 38976  LSHypclsh 38977  LPolclpoN 41483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-lpolN 41484

This theorem is referenced by: (None)