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Theorem dihmeetlem5 39821
Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihmeetlem5.b 𝐡 = (Baseβ€˜πΎ)
dihmeetlem5.l ≀ = (leβ€˜πΎ)
dihmeetlem5.j ∨ = (joinβ€˜πΎ)
dihmeetlem5.m ∧ = (meetβ€˜πΎ)
dihmeetlem5.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
dihmeetlem5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑄)) = ((𝑋 ∧ π‘Œ) ∨ 𝑄))

Proof of Theorem dihmeetlem5
StepHypRef Expression
1 simpl1 1192 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
2 simprl 770 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
3 simpl2 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
4 simpl3 1194 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ π‘Œ ∈ 𝐡)
5 simprr 772 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ≀ 𝑋)
6 dihmeetlem5.b . . . 4 𝐡 = (Baseβ€˜πΎ)
7 dihmeetlem5.l . . . 4 ≀ = (leβ€˜πΎ)
8 dihmeetlem5.j . . . 4 ∨ = (joinβ€˜πΎ)
9 dihmeetlem5.m . . . 4 ∧ = (meetβ€˜πΎ)
10 dihmeetlem5.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
116, 7, 8, 9, 10atmod2i1 38374 . . 3 ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑄 ≀ 𝑋) β†’ ((𝑋 ∧ π‘Œ) ∨ 𝑄) = (𝑋 ∧ (π‘Œ ∨ 𝑄)))
121, 2, 3, 4, 5, 11syl131anc 1384 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ ((𝑋 ∧ π‘Œ) ∨ 𝑄) = (𝑋 ∧ (π‘Œ ∨ 𝑄)))
1312eqcomd 2739 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑄)) = ((𝑋 ∧ π‘Œ) ∨ 𝑄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  lecple 17148  joincjn 18208  meetcmee 18209  Atomscatm 37775  HLchlt 37862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-lat 18329  df-clat 18396  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-psubsp 38016  df-pmap 38017  df-padd 38309
This theorem is referenced by:  dihmeetlem6  39822  dihjatc1  39824  dihmeetlem10N  39829
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