Theorem dihmeetlem17N 42065

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem14.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem14.l	⊢ ≤ = (le‘𝐾)
dihmeetlem14.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem14.j	⊢ ∨ = (join‘𝐾)
dihmeetlem14.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem14.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem14.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem14.s	⊢ ⊕ = (LSSum‘𝑈)
dihmeetlem14.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem17.o	⊢ 0 = (0.‘𝐾)

Assertion

Ref	Expression
dihmeetlem17N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∧ 𝑝) = 0 )

Proof of Theorem dihmeetlem17N

Step	Hyp	Ref	Expression
1		simpl1l 1241	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ HL)
2	1	hllatd 40106	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ Lat)
3		simpl3l 1245	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐴)
4		dihmeetlem14.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		dihmeetlem14.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 40031	. . . 4 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
7	3, 6	syl 18	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐵)
8		simpr1 1211	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
9		dihmeetlem14.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
10	4, 9	latmcom 18518	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∧ 𝑌) = (𝑌 ∧ 𝑝))
11	2, 7, 8, 10	syl3anc 1396	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑝 ∧ 𝑌) = (𝑌 ∧ 𝑝))
12		simpl1 1208	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		simpl2 1209	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
14		simpl3 1210	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
15		simpr2 1212	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑋 ∧ 𝑌) ≤ 𝑊)
16		simpr3 1213	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝑝 ≤ 𝑋)
17		dihmeetlem14.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
18		dihmeetlem14.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
19		dihmeetlem14.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
20	4, 17, 18, 9, 5, 19	lhpmcvr4N 40768	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → ¬ 𝑝 ≤ 𝑌)
21	12, 13, 14, 8, 15, 16, 20	syl123anc 1412	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → ¬ 𝑝 ≤ 𝑌)
22		hlatl 40102	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
23	1, 22	syl 18	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ AtLat)
24		dihmeetlem17.o	. . . . 5 ⊢ 0 = (0.‘𝐾)
25	4, 17, 9, 24, 5	atnle 40059	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = 0 ))
26	23, 3, 8, 25	syl3anc 1396	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = 0 ))
27	21, 26	mpbid 235	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑝 ∧ 𝑌) = 0 )
28	11, 27	eqtr3d 2798	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∧ 𝑝) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   class class class wbr 5108  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  lecple 17316  joincjn 18366  meetcmee 18367  0.cp0 18476  Latclat 18486  LSSumclsm 19703  Atomscatm 40005  AtLatcal 40006  HLchlt 40092  LHypclh 40726  DVecHcdvh 41820  DIsoHcdih 41970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-lat 18487  df-covers 40008  df-ats 40009  df-atl 40040  df-cvlat 40064  df-hlat 40093  df-lhyp 40730

This theorem is referenced by:  dihmeetlem18N  42066