Theorem dihmeetlem5 42032

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)

Hypotheses

Ref	Expression
dihmeetlem5.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem5.l	⊢ ≤ = (le‘𝐾)
dihmeetlem5.j	⊢ ∨ = (join‘𝐾)
dihmeetlem5.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem5.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
dihmeetlem5	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄))

Proof of Theorem dihmeetlem5

Step	Hyp	Ref	Expression
1		simpl1 1208	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simprl 782	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
3		simpl2 1209	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
4		simpl3 1210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
5		simprr 784	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ≤ 𝑋)
6		dihmeetlem5.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		dihmeetlem5.l	. . . 4 ⊢ ≤ = (le‘𝐾)
8		dihmeetlem5.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9		dihmeetlem5.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
10		dihmeetlem5.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11	6, 7, 8, 9, 10	atmod2i1 40585	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑄 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∨ 𝑄) = (𝑋 ∧ (𝑌 ∨ 𝑄)))
12	1, 2, 3, 4, 5, 11	syl131anc 1408	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → ((𝑋 ∧ 𝑌) ∨ 𝑄) = (𝑋 ∧ (𝑌 ∨ 𝑄)))
13	12	eqcomd 2776	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150   class class class wbr 5114  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  lecple 17320  joincjn 18370  meetcmee 18371  Atomscatm 39987  HLchlt 40074

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 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

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 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-proset 18353  df-poset 18372  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18491  df-clat 18558  df-oposet 39900  df-ol 39902  df-oml 39903  df-covers 39990  df-ats 39991  df-atl 40022  df-cvlat 40046  df-hlat 40075  df-psubsp 40227  df-pmap 40228  df-padd 40520

This theorem is referenced by:  dihmeetlem6  42033  dihjatc1  42035  dihmeetlem10N  42040