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Theorem lvolcmp 39122
Description: If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
lvolcmp.l ≀ = (leβ€˜πΎ)
lvolcmp.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvolcmp ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
Proof of Theorem lvolcmp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp2 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
2 simp1 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝐾 ∈ HL)
3 eqid 2728 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 lvolcmp.v . . . . . . 7 𝑉 = (LVolsβ€˜πΎ)
53, 4lvolbase 39083 . . . . . 6 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
653ad2ant2 1131 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
7 eqid 2728 . . . . . 6 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
8 eqid 2728 . . . . . 6 (LPlanesβ€˜πΎ) = (LPlanesβ€˜πΎ)
93, 7, 8, 4islvol4 39079 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ (𝑋 ∈ 𝑉 ↔ βˆƒπ‘§ ∈ (LPlanesβ€˜πΎ)𝑧( β‹– β€˜πΎ)𝑋))
102, 6, 9syl2anc 582 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ∈ 𝑉 ↔ βˆƒπ‘§ ∈ (LPlanesβ€˜πΎ)𝑧( β‹– β€˜πΎ)𝑋))
111, 10mpbid 231 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ βˆƒπ‘§ ∈ (LPlanesβ€˜πΎ)𝑧( β‹– β€˜πΎ)𝑋)
12 simpr3 1193 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 ≀ π‘Œ)
13 hlpos 38870 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
14133ad2ant1 1130 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝐾 ∈ Poset)
1514adantr 479 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝐾 ∈ Poset)
166adantr 479 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
17 simpl3 1190 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ π‘Œ ∈ 𝑉)
183, 4lvolbase 39083 . . . . . . . 8 (π‘Œ ∈ 𝑉 β†’ π‘Œ ∈ (Baseβ€˜πΎ))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ π‘Œ ∈ (Baseβ€˜πΎ))
20 simpr1 1191 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧 ∈ (LPlanesβ€˜πΎ))
213, 8lplnbase 39039 . . . . . . . 8 (𝑧 ∈ (LPlanesβ€˜πΎ) β†’ 𝑧 ∈ (Baseβ€˜πΎ))
2220, 21syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧 ∈ (Baseβ€˜πΎ))
23 simpr2 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧( β‹– β€˜πΎ)𝑋)
24 simpl1 1188 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝐾 ∈ HL)
25 lvolcmp.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
263, 25, 7cvrle 38782 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ∈ (Baseβ€˜πΎ)) ∧ 𝑧( β‹– β€˜πΎ)𝑋) β†’ 𝑧 ≀ 𝑋)
2724, 22, 16, 23, 26syl31anc 1370 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧 ≀ 𝑋)
283, 25postr 18319 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑧 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ))) β†’ ((𝑧 ≀ 𝑋 ∧ 𝑋 ≀ π‘Œ) β†’ 𝑧 ≀ π‘Œ))
2915, 22, 16, 19, 28syl13anc 1369 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ ((𝑧 ≀ 𝑋 ∧ 𝑋 ≀ π‘Œ) β†’ 𝑧 ≀ π‘Œ))
3027, 12, 29mp2and 697 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧 ≀ π‘Œ)
3125, 7, 8, 4lplncvrlvol2 39120 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑧 ∈ (LPlanesβ€˜πΎ) ∧ π‘Œ ∈ 𝑉) ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧( β‹– β€˜πΎ)π‘Œ)
3224, 20, 17, 30, 31syl31anc 1370 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑧( β‹– β€˜πΎ)π‘Œ)
333, 25, 7cvrcmp 38787 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) ∧ (𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑧( β‹– β€˜πΎ)π‘Œ)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
3415, 16, 19, 22, 23, 32, 33syl132anc 1385 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
3512, 34mpbid 231 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (𝑧 ∈ (LPlanesβ€˜πΎ) ∧ 𝑧( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 = π‘Œ)
36353exp2 1351 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑧 ∈ (LPlanesβ€˜πΎ) β†’ (𝑧( β‹– β€˜πΎ)𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ))))
3736rexlimdv 3150 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (βˆƒπ‘§ ∈ (LPlanesβ€˜πΎ)𝑧( β‹– β€˜πΎ)𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ)))
3811, 37mpd 15 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ))
393, 25posref 18317 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ 𝑋 ≀ 𝑋)
4014, 6, 39syl2anc 582 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ≀ 𝑋)
41 breq2 5156 . . 3 (𝑋 = π‘Œ β†’ (𝑋 ≀ 𝑋 ↔ 𝑋 ≀ π‘Œ))
4240, 41syl5ibcom 244 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 = π‘Œ β†’ 𝑋 ≀ π‘Œ))
4338, 42impbid 211 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  Basecbs 17187  lecple 17247  Posetcpo 18306   β‹– ccvr 38766  HLchlt 38854  LPlanesclpl 38997  LVolsclvol 38998
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005
This theorem is referenced by:  lvolnltN  39123  2lplnja  39124
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