Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihmeetlem5 Structured version   Visualization version   GIF version

Theorem dihmeetlem5 37116
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 ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → (𝑋 (𝑌 𝑄)) = ((𝑋 𝑌) 𝑄))

Proof of Theorem dihmeetlem5
StepHypRef Expression
1 simpl1 1227 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → 𝐾 ∈ HL)
2 simprl 754 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → 𝑄𝐴)
3 simpl2 1229 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → 𝑋𝐵)
4 simpl3 1231 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → 𝑌𝐵)
5 simprr 756 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 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)
116, 7, 8, 9, 10atmod2i1 35668 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑋𝐵𝑌𝐵) ∧ 𝑄 𝑋) → ((𝑋 𝑌) 𝑄) = (𝑋 (𝑌 𝑄)))
121, 2, 3, 4, 5, 11syl131anc 1489 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → ((𝑋 𝑌) 𝑄) = (𝑋 (𝑌 𝑄)))
1312eqcomd 2777 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → (𝑋 (𝑌 𝑄)) = ((𝑋 𝑌) 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35070  HLchlt 35157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-psubsp 35310  df-pmap 35311  df-padd 35603
This theorem is referenced by:  dihmeetlem6  37117  dihjatc1  37119  dihmeetlem10N  37124
  Copyright terms: Public domain W3C validator