Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omlmod1i2N Structured version   Visualization version   GIF version

Theorem omlmod1i2N 35067
Description: Analogue of modular law atmod1i2 35666 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlmod.b 𝐵 = (Base‘𝐾)
omlmod.l = (le‘𝐾)
omlmod.j = (join‘𝐾)
omlmod.m = (meet‘𝐾)
omlmod.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
omlmod1i2N ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))

Proof of Theorem omlmod1i2N
StepHypRef Expression
1 simp1 1130 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OML)
2 simp23 1250 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
3 simp21 1248 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
4 simp22 1249 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 simp3l 1243 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋 𝑍)
6 omlmod.b . . . . . . 7 𝐵 = (Base‘𝐾)
7 omlmod.l . . . . . . 7 = (le‘𝐾)
8 omlmod.c . . . . . . 7 𝐶 = (cm‘𝐾)
96, 7, 8lecmtN 35063 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍𝑋𝐶𝑍))
101, 3, 2, 9syl3anc 1476 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍𝑋𝐶𝑍))
115, 10mpd 15 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑋𝐶𝑍)
126, 8cmtcomN 35056 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑍𝐶𝑋))
131, 3, 2, 12syl3anc 1476 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋𝐶𝑍𝑍𝐶𝑋))
1411, 13mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑋)
15 simp3r 1244 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑌𝐶𝑍)
166, 8cmtcomN 35056 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑍𝐶𝑌))
171, 4, 2, 16syl3anc 1476 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑌𝐶𝑍𝑍𝐶𝑌))
1815, 17mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝑍𝐶𝑌)
19 omlmod.j . . . 4 = (join‘𝐾)
20 omlmod.m . . . 4 = (meet‘𝐾)
216, 19, 20, 8omlfh1N 35065 . . 3 ((𝐾 ∈ OML ∧ (𝑍𝐵𝑋𝐵𝑌𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
221, 2, 3, 4, 14, 18, 21syl132anc 1494 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑍 𝑋) (𝑍 𝑌)))
23 omllat 35049 . . . 4 (𝐾 ∈ OML → 𝐾 ∈ Lat)
24233ad2ant1 1127 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Lat)
256, 19latjcl 17259 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2624, 3, 4, 25syl3anc 1476 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑌) ∈ 𝐵)
276, 20latmcom 17283 . . 3 ((𝐾 ∈ Lat ∧ 𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
2824, 2, 26, 27syl3anc 1476 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 (𝑋 𝑌)) = ((𝑋 𝑌) 𝑍))
296, 7, 20latleeqm2 17288 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
3024, 3, 2, 29syl3anc 1476 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 𝑍 ↔ (𝑍 𝑋) = 𝑋))
315, 30mpbid 222 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑋) = 𝑋)
326, 20latmcom 17283 . . . 4 ((𝐾 ∈ Lat ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) = (𝑌 𝑍))
3324, 2, 4, 32syl3anc 1476 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑍 𝑌) = (𝑌 𝑍))
3431, 33oveq12d 6814 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → ((𝑍 𝑋) (𝑍 𝑌)) = (𝑋 (𝑌 𝑍)))
3522, 28, 343eqtr3rd 2814 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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  Latclat 17253  cmccmtN 34980  OMLcoml 34982
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-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-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-oposet 34983  df-cmtN 34984  df-ol 34985  df-oml 34986
This theorem is referenced by:  omlspjN  35068
  Copyright terms: Public domain W3C validator