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Theorem lvolcmp 38476
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 1137 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
2 simp1 1136 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝐾 ∈ HL)
3 eqid 2732 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 lvolcmp.v . . . . . . 7 𝑉 = (LVolsβ€˜πΎ)
53, 4lvolbase 38437 . . . . . 6 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
653ad2ant2 1134 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
7 eqid 2732 . . . . . 6 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
8 eqid 2732 . . . . . 6 (LPlanesβ€˜πΎ) = (LPlanesβ€˜πΎ)
93, 7, 8, 4islvol4 38433 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ (𝑋 ∈ 𝑉 ↔ βˆƒπ‘§ ∈ (LPlanesβ€˜πΎ)𝑧( β‹– β€˜πΎ)𝑋))
102, 6, 9syl2anc 584 . . . 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 1196 . . . . . 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 38224 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
14133ad2ant1 1133 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝐾 ∈ Poset)
1514adantr 481 . . . . . . 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 481 . . . . . . 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 1193 . . . . . . . 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 38437 . . . . . . . 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 1194 . . . . . . . 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 38393 . . . . . . . 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 1195 . . . . . . 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 1191 . . . . . . . 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 38136 . . . . . . . . . 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 1373 . . . . . . . . 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 18269 . . . . . . . . . 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 1372 . . . . . . . . 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 38474 . . . . . . . 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 1373 . . . . . . 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 38141 . . . . . . 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 1388 . . . . . 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 1354 . . . 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 3153 . . 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 18267 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ 𝑋 ≀ 𝑋)
4014, 6, 39syl2anc 584 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ≀ 𝑋)
41 breq2 5151 . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   class class class wbr 5147  β€˜cfv 6540  Basecbs 17140  lecple 17200  Posetcpo 18256   β‹– ccvr 38120  HLchlt 38208  LPlanesclpl 38351  LVolsclvol 38352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lvols 38359
This theorem is referenced by:  lvolnltN  38477  2lplnja  38478
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