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Theorem isoml 35047
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐵 = (Base‘𝐾)
isoml.l = (le‘𝐾)
isoml.j = (join‘𝐾)
isoml.m = (meet‘𝐾)
isoml.o = (oc‘𝐾)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isoml
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6333 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isoml.b . . . 4 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2823 . . 3 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6333 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 isoml.l . . . . . . 7 = (le‘𝐾)
64, 5syl6eqr 2823 . . . . . 6 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 4798 . . . . 5 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
8 fveq2 6333 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 isoml.j . . . . . . . 8 = (join‘𝐾)
108, 9syl6eqr 2823 . . . . . . 7 (𝑘 = 𝐾 → (join‘𝑘) = )
11 eqidd 2772 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
12 fveq2 6333 . . . . . . . . 9 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13 isoml.m . . . . . . . . 9 = (meet‘𝐾)
1412, 13syl6eqr 2823 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = )
15 eqidd 2772 . . . . . . . 8 (𝑘 = 𝐾𝑦 = 𝑦)
16 fveq2 6333 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 isoml.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17syl6eqr 2823 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
1918fveq1d 6335 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( 𝑥))
2014, 15, 19oveq123d 6817 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ( 𝑥)))
2110, 11, 20oveq123d 6817 . . . . . 6 (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 (𝑦 ( 𝑥))))
2221eqeq2d 2781 . . . . 5 (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 (𝑦 ( 𝑥)))))
237, 22imbi12d 333 . . . 4 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
243, 23raleqbidv 3301 . . 3 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
253, 24raleqbidv 3301 . 2 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
26 df-oml 34988 . 2 OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
2725, 26elrab2 3518 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  occoc 16157  joincjn 17152  meetcmee 17153  OLcol 34983  OMLcoml 34984
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
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-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-oml 34988
This theorem is referenced by:  isomliN  35048  omlol  35049  omllaw  35052
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