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Theorem atcvrj0 39410
Description: Two atoms covering the zero subspace are equal. (atcv1 32408 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 5150 . . . . . . . 8 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) ↔ 0 𝐶(𝑃 𝑄)))
21biimpd 229 . . . . . . 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 39408 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1181 . . . . . . 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 239 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄))
1211ex 412 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄)))
1312com23 86 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑋 = 0𝑃 = 𝑄)))
14133impia 1116 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
15 oveq1 7437 . . . . . . 7 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1615breq2d 5159 . . . . . 6 (𝑃 = 𝑄 → (𝑋𝐶(𝑃 𝑄) ↔ 𝑋𝐶(𝑄 𝑄)))
1716biimpac 478 . . . . 5 ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 𝑄))
18 simpr3 1195 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
194, 7hlatjidm 39350 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
2018, 19syldan 591 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑄) = 𝑄)
2120breq2d 5159 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) ↔ 𝑋𝐶𝑄))
22 hlatl 39341 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2322adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
24 simpr1 1193 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
25 atcvrj0.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
26 eqid 2734 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
2725, 26, 5, 6, 7atcvreq0 39295 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑄𝐴) → (𝑋𝐶𝑄𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1370 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
2928biimpd 229 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
3021, 29sylbid 240 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) → 𝑋 = 0 ))
3117, 30syl5 34 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
3231expd 415 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑃 = 𝑄𝑋 = 0 )))
33323impia 1116 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑃 = 𝑄𝑋 = 0 ))
3414, 33impbid 212 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  0.cp0 18480  ccvr 39243  Atomscatm 39244  AtLatcal 39245  HLchlt 39331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332
This theorem is referenced by:  cvrat2  39411  atcvrneN  39412  atcvrj2b  39414
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