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Theorem atcvrj0 33732
 Description: Two atoms covering the zero subspace are equal. (atcv1 28623 analog.) (Contributed by NM, 29-Nov-2011.)
Hypotheses
Ref Expression
atcvrj0.b 𝐵 = (Base‘𝐾)
atcvrj0.j = (join‘𝐾)
atcvrj0.z 0 = (0.‘𝐾)
atcvrj0.c 𝐶 = ( ⋖ ‘𝐾)
atcvrj0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atcvrj0 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))

Proof of Theorem atcvrj0
StepHypRef Expression
1 breq1 4586 . . . . . . . 8 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) ↔ 0 𝐶(𝑃 𝑄)))
21biimpd 218 . . . . . . 7 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 0 𝐶(𝑃 𝑄)))
32adantl 481 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 𝑄) → 0 𝐶(𝑃 𝑄)))
4 atcvrj0.j . . . . . . . . 9 = (join‘𝐾)
5 atcvrj0.z . . . . . . . . 9 0 = (0.‘𝐾)
6 atcvrj0.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
7 atcvrj0.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
84, 5, 6, 7atcvr0eq 33730 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1266 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
109adantr 480 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
113, 10sylibd 228 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄))
1211ex 449 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄)))
1312com23 84 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑋 = 0𝑃 = 𝑄)))
14133impia 1253 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
15 oveq1 6556 . . . . . . 7 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1615breq2d 4595 . . . . . 6 (𝑃 = 𝑄 → (𝑋𝐶(𝑃 𝑄) ↔ 𝑋𝐶(𝑄 𝑄)))
1716biimpac 502 . . . . 5 ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 𝑄))
18 simpr3 1062 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
194, 7hlatjidm 33673 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
2018, 19syldan 486 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑄) = 𝑄)
2120breq2d 4595 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) ↔ 𝑋𝐶𝑄))
22 hlatl 33665 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2322adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
24 simpr1 1060 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
25 atcvrj0.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
26 eqid 2610 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
2725, 26, 5, 6, 7atcvreq0 33619 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑄𝐴) → (𝑋𝐶𝑄𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1318 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
2928biimpd 218 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
3021, 29sylbid 229 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) → 𝑋 = 0 ))
3117, 30syl5 33 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
3231expd 451 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑃 = 𝑄𝑋 = 0 )))
33323impia 1253 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑃 = 𝑄𝑋 = 0 ))
3414, 33impbid 201 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  0.cp0 16860   ⋖ ccvr 33567  Atomscatm 33568  AtLatcal 33569  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  cvrat2  33733  atcvrneN  33734  atcvrj2b  33736
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