Theorem pexmidlem6N 35261

Description: Lemma for pexmidN 35255. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pexmidlem.l	⊢ ≤ = (le‘𝐾)
pexmidlem.j	⊢ ∨ = (join‘𝐾)
pexmidlem.a	⊢ 𝐴 = (Atoms‘𝐾)
pexmidlem.p	⊢ + = (+𝑃‘𝐾)
pexmidlem.o	⊢ ⊥ = (⊥𝑃‘𝐾)
pexmidlem.m	⊢ 𝑀 = (𝑋 + {𝑝})

Assertion

Ref	Expression
pexmidlem6N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 = 𝑋)

Proof of Theorem pexmidlem6N

Step	Hyp	Ref	Expression
1		pexmidlem.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
2		pexmidlem.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		pexmidlem.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
4		pexmidlem.p	. . . . . . . 8 ⊢ + = (+𝑃‘𝐾)
5		pexmidlem.o	. . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾)
6		pexmidlem.m	. . . . . . . 8 ⊢ 𝑀 = (𝑋 + {𝑝})
7	1, 2, 3, 4, 5, 6	pexmidlem5N 35260	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅)
8	7	3adantr1 1220	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅)
9	8	fveq2d 6195	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘∅))
10		simpl1 1064	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝐾 ∈ HL)
11	3, 5	pol0N 35195	. . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
12	10, 11	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘∅) = 𝐴)
13	9, 12	eqtrd 2656	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝐴)
14	13	ineq1d 3813	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝐴 ∩ 𝑀))
15		simpl2 1065	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ⊆ 𝐴)
16		simpl3 1066	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑝 ∈ 𝐴)
17	16	snssd 4340	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → {𝑝} ⊆ 𝐴)
18	3, 4	paddssat 35100	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ {𝑝} ⊆ 𝐴) → (𝑋 + {𝑝}) ⊆ 𝐴)
19	10, 15, 17, 18	syl3anc 1326	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝑋 + {𝑝}) ⊆ 𝐴)
20	6, 19	syl5eqss 3649	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ⊆ 𝐴)
21	10, 15, 20	3jca 1242	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑀 ⊆ 𝐴))
22	3, 4	sspadd1 35101	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ {𝑝} ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + {𝑝}))
23	10, 15, 17, 22	syl3anc 1326	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ⊆ (𝑋 + {𝑝}))
24	23, 6	syl6sseqr 3652	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ⊆ 𝑀)
25		simpr1 1067	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
26		eqid 2622	. . . . . . . . . . 11 ⊢ (PSubCl‘𝐾) = (PSubCl‘𝐾)
27	3, 5, 26	ispsubclN 35223	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
28	10, 27	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
29	15, 25, 28	mpbir2and 957	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑋 ∈ (PSubCl‘𝐾))
30	3, 4, 26	paddatclN 35235	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ 𝑝 ∈ 𝐴) → (𝑋 + {𝑝}) ∈ (PSubCl‘𝐾))
31	10, 29, 16, 30	syl3anc 1326	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝑋 + {𝑝}) ∈ (PSubCl‘𝐾))
32	6, 31	syl5eqel 2705	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ∈ (PSubCl‘𝐾))
33	5, 26	psubcli2N 35225	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑀 ∈ (PSubCl‘𝐾)) → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
34	10, 32, 33	syl2anc 693	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
35	24, 34	jca 554	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝑋 ⊆ 𝑀 ∧ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
36	3, 5	poml4N 35239	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑀 ⊆ 𝐴) → ((𝑋 ⊆ 𝑀 ∧ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋))))
37	21, 35, 36	sylc 65	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
38		sseqin2 3817	. . . 4 ⊢ (𝑀 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑀) = 𝑀)
39	20, 38	sylib 208	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (𝐴 ∩ 𝑀) = 𝑀)
40	14, 37, 39	3eqtr3rd 2665	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
41	40, 25	eqtrd 2656	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ‘cfv 5888  (class class class)co 6650  lecple 15948  joincjn 16944  Atomscatm 34550  HLchlt 34637  +_𝑃cpadd 35081  ⊥_𝑃cpolN 35188  PSubClcpscN 35220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  pexmidlem8N  35263