Theorem lhpj1 39721

Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unity. (Contributed by NM, 7-Dec-2012.)

Hypotheses

Ref	Expression
lhpj1.b	⊢ 𝐵 = (Base‘𝐾)
lhpj1.l	⊢ ≤ = (le‘𝐾)
lhpj1.j	⊢ ∨ = (join‘𝐾)
lhpj1.u	⊢ 1 = (1.‘𝐾)
lhpj1.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhpj1	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )

Proof of Theorem lhpj1

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 765	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ HL)
2		simpr 483	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
3		lhpj1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
4		lhpj1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5	3, 4	lhpbase 39697	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
6	5	ad2antlr 725	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ 𝐵)
7		lhpj1.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		eqid 2726	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
9	3, 7, 8	hlrelat2 39102	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)))
10	1, 2, 6, 9	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)))
11		simp1l 1194	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp2 1134	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13		simp3r 1199	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
14		lhpj1.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
15		lhpj1.u	. . . . . . . 8 ⊢ 1 = (1.‘𝐾)
16	7, 14, 15, 8, 4	lhpjat1 39719	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
17	11, 12, 13, 16	syl12anc 835	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
18		simp3l 1198	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ≤ 𝑋)
19		simp1ll 1233	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ HL)
20	19	hllatd 39062	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ Lat)
21	3, 8	atbase 38987	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
22	21	3ad2ant2 1131	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ 𝐵)
23		simp1r 1195	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
24	6	3ad2ant1 1130	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
25	3, 7, 14	latjlej2 18479	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑝 ≤ 𝑋 → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋)))
26	20, 22, 23, 24, 25	syl13anc 1369	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑝 ≤ 𝑋 → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋)))
27	18, 26	mpd 15	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋))
28	17, 27	eqbrtrrd 5177	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 1 ≤ (𝑊 ∨ 𝑋))
29		hlop 39060	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
30	19, 29	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ OP)
31	3, 14	latjcl 18464	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑊 ∨ 𝑋) ∈ 𝐵)
32	20, 24, 23, 31	syl3anc 1368	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑋) ∈ 𝐵)
33	3, 7, 15	op1le 38890	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (𝑊 ∨ 𝑋) ∈ 𝐵) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
34	30, 32, 33	syl2anc 582	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
35	28, 34	mpbid 231	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )
36	35	rexlimdv3a 3149	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊) → (𝑊 ∨ 𝑋) = 1 ))
37	10, 36	sylbid 239	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 → (𝑊 ∨ 𝑋) = 1 ))
38	37	impr 453	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   class class class wbr 5153  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  1.cp1 18449  Latclat 18456  OPcops 38870  Atomscatm 38961  HLchlt 39048  LHypclh 39683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-lhyp 39687

This theorem is referenced by:  lhpmcvr  39722  cdleme30a  40077