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Theorem oldmm1 36220
Description: De Morgan's law for meet in an ortholattice. (chdmm1 29216 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
oldmm1.b 𝐵 = (Base‘𝐾)
oldmm1.j = (join‘𝐾)
oldmm1.m = (meet‘𝐾)
oldmm1.o = (oc‘𝐾)
Assertion
Ref Expression
oldmm1 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))

Proof of Theorem oldmm1
StepHypRef Expression
1 oldmm1.b . 2 𝐵 = (Base‘𝐾)
2 eqid 2826 . 2 (le‘𝐾) = (le‘𝐾)
3 ollat 36216 . . 3 (𝐾 ∈ OL → 𝐾 ∈ Lat)
433ad2ant1 1127 . 2 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
5 olop 36217 . . . 4 (𝐾 ∈ OL → 𝐾 ∈ OP)
653ad2ant1 1127 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
7 oldmm1.m . . . . 5 = (meet‘𝐾)
81, 7latmcl 17652 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 8syl3an1 1157 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
10 oldmm1.o . . . 4 = (oc‘𝐾)
111, 10opoccl 36197 . . 3 ((𝐾 ∈ OP ∧ (𝑋 𝑌) ∈ 𝐵) → ( ‘(𝑋 𝑌)) ∈ 𝐵)
126, 9, 11syl2anc 584 . 2 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) ∈ 𝐵)
131, 10opoccl 36197 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
145, 13sylan 580 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
15143adant3 1126 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
161, 10opoccl 36197 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
175, 16sylan 580 . . . 4 ((𝐾 ∈ OL ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
18173adant2 1125 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
19 oldmm1.j . . . 4 = (join‘𝐾)
201, 19latjcl 17651 . . 3 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → (( 𝑋) ( 𝑌)) ∈ 𝐵)
214, 15, 18, 20syl3anc 1365 . 2 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ( 𝑌)) ∈ 𝐵)
221, 2, 19latlej1 17660 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → ( 𝑋)(le‘𝐾)(( 𝑋) ( 𝑌)))
234, 15, 18, 22syl3anc 1365 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋)(le‘𝐾)(( 𝑋) ( 𝑌)))
24 simp2 1131 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
251, 2, 10oplecon1b 36204 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ (( 𝑋) ( 𝑌)) ∈ 𝐵) → (( 𝑋)(le‘𝐾)(( 𝑋) ( 𝑌)) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑋))
266, 24, 21, 25syl3anc 1365 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋)(le‘𝐾)(( 𝑋) ( 𝑌)) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑋))
2723, 26mpbid 233 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑋)
281, 2, 19latlej2 17661 . . . . . 6 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → ( 𝑌)(le‘𝐾)(( 𝑋) ( 𝑌)))
294, 15, 18, 28syl3anc 1365 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌)(le‘𝐾)(( 𝑋) ( 𝑌)))
30 simp3 1132 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
311, 2, 10oplecon1b 36204 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑌𝐵 ∧ (( 𝑋) ( 𝑌)) ∈ 𝐵) → (( 𝑌)(le‘𝐾)(( 𝑋) ( 𝑌)) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑌))
326, 30, 21, 31syl3anc 1365 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌)(le‘𝐾)(( 𝑋) ( 𝑌)) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑌))
3329, 32mpbid 233 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑌)
341, 10opoccl 36197 . . . . . 6 ((𝐾 ∈ OP ∧ (( 𝑋) ( 𝑌)) ∈ 𝐵) → ( ‘(( 𝑋) ( 𝑌))) ∈ 𝐵)
356, 21, 34syl2anc 584 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) ∈ 𝐵)
361, 2, 7latlem12 17678 . . . . 5 ((𝐾 ∈ Lat ∧ (( ‘(( 𝑋) ( 𝑌))) ∈ 𝐵𝑋𝐵𝑌𝐵)) → ((( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑋 ∧ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑌) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)(𝑋 𝑌)))
374, 35, 24, 30, 36syl13anc 1366 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑋 ∧ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)𝑌) ↔ ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)(𝑋 𝑌)))
3827, 33, 37mpbi2and 708 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)(𝑋 𝑌))
391, 2, 10oplecon1b 36204 . . . 4 ((𝐾 ∈ OP ∧ (( 𝑋) ( 𝑌)) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ ( ‘(𝑋 𝑌))(le‘𝐾)(( 𝑋) ( 𝑌))))
406, 21, 9, 39syl3anc 1365 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (( ‘(( 𝑋) ( 𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ ( ‘(𝑋 𝑌))(le‘𝐾)(( 𝑋) ( 𝑌))))
4138, 40mpbid 233 . 2 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌))(le‘𝐾)(( 𝑋) ( 𝑌)))
421, 2, 7latmle1 17676 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
433, 42syl3an1 1157 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
441, 2, 10oplecon3b 36203 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ ( 𝑋)(le‘𝐾)( ‘(𝑋 𝑌))))
456, 9, 24, 44syl3anc 1365 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ ( 𝑋)(le‘𝐾)( ‘(𝑋 𝑌))))
4643, 45mpbid 233 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋)(le‘𝐾)( ‘(𝑋 𝑌)))
471, 2, 7latmle2 17677 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
483, 47syl3an1 1157 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
491, 2, 10oplecon3b 36203 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ ( 𝑌)(le‘𝐾)( ‘(𝑋 𝑌))))
506, 9, 30, 49syl3anc 1365 . . . 4 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ ( 𝑌)(le‘𝐾)( ‘(𝑋 𝑌))))
5148, 50mpbid 233 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌)(le‘𝐾)( ‘(𝑋 𝑌)))
521, 2, 19latjle12 17662 . . . 4 ((𝐾 ∈ Lat ∧ (( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵 ∧ ( ‘(𝑋 𝑌)) ∈ 𝐵)) → ((( 𝑋)(le‘𝐾)( ‘(𝑋 𝑌)) ∧ ( 𝑌)(le‘𝐾)( ‘(𝑋 𝑌))) ↔ (( 𝑋) ( 𝑌))(le‘𝐾)( ‘(𝑋 𝑌))))
534, 15, 18, 12, 52syl13anc 1366 . . 3 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋)(le‘𝐾)( ‘(𝑋 𝑌)) ∧ ( 𝑌)(le‘𝐾)( ‘(𝑋 𝑌))) ↔ (( 𝑋) ( 𝑌))(le‘𝐾)( ‘(𝑋 𝑌))))
5446, 51, 53mpbi2and 708 . 2 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ( 𝑌))(le‘𝐾)( ‘(𝑋 𝑌)))
551, 2, 4, 12, 21, 41, 54latasymd 17657 1 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  occoc 16563  joincjn 17544  meetcmee 17545  Latclat 17645  OPcops 36175  OLcol 36177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17528  df-poset 17546  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-lat 17646  df-oposet 36179  df-ol 36181
This theorem is referenced by:  oldmm2  36221  oldmm3N  36222  cmtcomlemN  36251  cmtbr2N  36256  omlfh1N  36261  cvrexch  36423  lhpmod2i2  37041  lhpmod6i1  37042  doca2N  38129  djajN  38140
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