Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omllaw Structured version   Unicode version

Theorem omllaw 29978
 Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b
omllaw.l
omllaw.j
omllaw.m
omllaw.o
Assertion
Ref Expression
omllaw

Proof of Theorem omllaw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5
2 omllaw.l . . . . 5
3 omllaw.j . . . . 5
4 omllaw.m . . . . 5
5 omllaw.o . . . . 5
61, 2, 3, 4, 5isoml 29973 . . . 4
76simprbi 451 . . 3
8 breq1 4207 . . . . 5
9 id 20 . . . . . . 7
10 fveq2 5720 . . . . . . . 8
1110oveq2d 6089 . . . . . . 7
129, 11oveq12d 6091 . . . . . 6
1312eqeq2d 2446 . . . . 5
148, 13imbi12d 312 . . . 4
15 breq2 4208 . . . . 5
16 id 20 . . . . . 6
17 oveq1 6080 . . . . . . 7
1817oveq2d 6089 . . . . . 6
1916, 18eqeq12d 2449 . . . . 5
2015, 19imbi12d 312 . . . 4
2114, 20rspc2v 3050 . . 3
227, 21syl5com 28 . 2
23223impib 1151 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697   class class class wbr 4204  cfv 5446  (class class class)co 6073  cbs 13461  cple 13528  coc 13529  cjn 14393  cmee 14394  col 29909  coml 29910 This theorem is referenced by:  omllaw2N  29979  omllaw3  29980  omllaw4  29981 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-oml 29914
 Copyright terms: Public domain W3C validator