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Theorem atcvrj0 36686
Description: Two atoms covering the zero subspace are equal. (atcv1 30161 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 5045 . . . . . . . 8 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) ↔ 0 𝐶(𝑃 𝑄)))
21biimpd 232 . . . . . . 7 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 0 𝐶(𝑃 𝑄)))
32adantl 485 . . . . . 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 36684 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1179 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
109adantr 484 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
113, 10sylibd 242 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄))
1211ex 416 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄)))
1312com23 86 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑋 = 0𝑃 = 𝑄)))
14133impia 1114 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
15 oveq1 7147 . . . . . . 7 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1615breq2d 5054 . . . . . 6 (𝑃 = 𝑄 → (𝑋𝐶(𝑃 𝑄) ↔ 𝑋𝐶(𝑄 𝑄)))
1716biimpac 482 . . . . 5 ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 𝑄))
18 simpr3 1193 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
194, 7hlatjidm 36627 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
2018, 19syldan 594 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑄) = 𝑄)
2120breq2d 5054 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) ↔ 𝑋𝐶𝑄))
22 hlatl 36618 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2322adantr 484 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
24 simpr1 1191 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
25 atcvrj0.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
26 eqid 2822 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
2725, 26, 5, 6, 7atcvreq0 36572 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑄𝐴) → (𝑋𝐶𝑄𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
2928biimpd 232 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
3021, 29sylbid 243 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) → 𝑋 = 0 ))
3117, 30syl5 34 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
3231expd 419 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑃 = 𝑄𝑋 = 0 )))
33323impia 1114 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑃 = 𝑄𝑋 = 0 ))
3414, 33impbid 215 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  0.cp0 17638  ccvr 36520  Atomscatm 36521  AtLatcal 36522  HLchlt 36608
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-lat 17647  df-clat 17709  df-oposet 36434  df-ol 36436  df-oml 36437  df-covers 36524  df-ats 36525  df-atl 36556  df-cvlat 36580  df-hlat 36609
This theorem is referenced by:  cvrat2  36687  atcvrneN  36688  atcvrj2b  36690
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