Theorem dihmeetlem5 41251

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 1191	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simprl 770	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
3		simpl2 1192	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
4		simpl3 1193	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
5		simprr 772	. . 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 39804	. . 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 1384	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → ((𝑋 ∧ 𝑌) ∨ 𝑄) = (𝑋 ∧ (𝑌 ∨ 𝑄)))
13	12	eqcomd 2740	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5125  ‘cfv 6542  (class class class)co 7414  Basecbs 17230  lecple 17284  joincjn 18332  meetcmee 18333  Atomscatm 39205  HLchlt 39292

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 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-iin 4976  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-proset 18315  df-poset 18334  df-plt 18349  df-lub 18365  df-glb 18366  df-join 18367  df-meet 18368  df-p0 18444  df-lat 18451  df-clat 18518  df-oposet 39118  df-ol 39120  df-oml 39121  df-covers 39208  df-ats 39209  df-atl 39240  df-cvlat 39264  df-hlat 39293  df-psubsp 39446  df-pmap 39447  df-padd 39739

This theorem is referenced by:  dihmeetlem6  41252  dihjatc1  41254  dihmeetlem10N  41259