Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atcvrj0 Structured version   Visualization version   GIF version
Theorem atcvrj0 38602
Description: Two atoms covering the zero subspace are equal. (atcv1 31888 analog.) (Contributed by NM, 29-Nov-2011.)
Hypotheses
Ref Expression
atcvrj0.b 𝐡 = (Baseβ€˜πΎ)
atcvrj0.j ∨ = (joinβ€˜πΎ)
atcvrj0.z 0 = (0.β€˜πΎ)
atcvrj0.c 𝐢 = ( β‹– β€˜πΎ)
atcvrj0.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atcvrj0 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = 0 ↔ 𝑃 = 𝑄))
Proof of Theorem atcvrj0
StepHypRef Expression
1 breq1 5151 . . . . . . . 8 (𝑋 = 0 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) ↔ 0 𝐢(𝑃 ∨ 𝑄)))
21biimpd 228 . . . . . . 7 (𝑋 = 0 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 0 𝐢(𝑃 ∨ 𝑄)))
32adantl 482 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 0 𝐢(𝑃 ∨ 𝑄)))
4 atcvrj0.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
5 atcvrj0.z . . . . . . . . 9 0 = (0.β€˜πΎ)
6 atcvrj0.c . . . . . . . . 9 𝐢 = ( β‹– β€˜πΎ)
7 atcvrj0.a . . . . . . . . 9 𝐴 = (Atomsβ€˜πΎ)
84, 5, 6, 7atcvr0eq 38600 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1182 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
109adantr 481 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
113, 10sylibd 238 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑃 = 𝑄))
1211ex 413 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 = 0 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑃 = 𝑄)))
1312com23 86 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑋 = 0 β†’ 𝑃 = 𝑄)))
14133impia 1117 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = 0 β†’ 𝑃 = 𝑄))
15 oveq1 7418 . . . . . . 7 (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
1615breq2d 5160 . . . . . 6 (𝑃 = 𝑄 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) ↔ 𝑋𝐢(𝑄 ∨ 𝑄)))
1716biimpac 479 . . . . 5 ((𝑋𝐢(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) β†’ 𝑋𝐢(𝑄 ∨ 𝑄))
18 simpr3 1196 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
194, 7hlatjidm 38542 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) β†’ (𝑄 ∨ 𝑄) = 𝑄)
2018, 19syldan 591 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄 ∨ 𝑄) = 𝑄)
2120breq2d 5160 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑄 ∨ 𝑄) ↔ 𝑋𝐢𝑄))
22 hlatl 38533 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
2322adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ AtLat)
24 simpr1 1194 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
25 atcvrj0.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
26 eqid 2732 . . . . . . . . 9 (leβ€˜πΎ) = (leβ€˜πΎ)
2725, 26, 5, 6, 7atcvreq0 38487 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋𝐢𝑄 ↔ 𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1371 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢𝑄 ↔ 𝑋 = 0 ))
2928biimpd 228 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢𝑄 β†’ 𝑋 = 0 ))
3021, 29sylbid 239 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑄 ∨ 𝑄) β†’ 𝑋 = 0 ))
3117, 30syl5 34 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋𝐢(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) β†’ 𝑋 = 0 ))
3231expd 416 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑃 = 𝑄 β†’ 𝑋 = 0 )))
33323impia 1117 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑃 = 𝑄 β†’ 𝑋 = 0 ))
3414, 33impbid 211 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = 0 ↔ 𝑃 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  joincjn 18268  0.cp0 18380   β‹– ccvr 38435  Atomscatm 38436  AtLatcal 38437  HLchlt 38523
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524
This theorem is referenced by:  cvrat2  38603  atcvrneN  38604  atcvrj2b  38606
  Copyright terms: Public domain W3C validator