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Theorem atcvrj0 33535
Description: Two atoms covering the zero subspace are equal. (atcv1 28429 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 4580 . . . . . . . 8 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) ↔ 0 𝐶(𝑃 𝑄)))
21biimpd 217 . . . . . . 7 (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 0 𝐶(𝑃 𝑄)))
32adantl 480 . . . . . 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 33533 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1265 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
109adantr 479 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))
113, 10sylibd 227 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄))
1211ex 448 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 𝑄) → 𝑃 = 𝑄)))
1312com23 83 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑋 = 0𝑃 = 𝑄)))
14133impia 1252 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
15 oveq1 6534 . . . . . . 7 (𝑃 = 𝑄 → (𝑃 𝑄) = (𝑄 𝑄))
1615breq2d 4589 . . . . . 6 (𝑃 = 𝑄 → (𝑋𝐶(𝑃 𝑄) ↔ 𝑋𝐶(𝑄 𝑄)))
1716biimpac 501 . . . . 5 ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 𝑄))
18 simpr3 1061 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
194, 7hlatjidm 33476 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
2018, 19syldan 485 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑄) = 𝑄)
2120breq2d 4589 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) ↔ 𝑋𝐶𝑄))
22 hlatl 33468 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2322adantr 479 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
24 simpr1 1059 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
25 atcvrj0.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
26 eqid 2609 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
2725, 26, 5, 6, 7atcvreq0 33422 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑄𝐴) → (𝑋𝐶𝑄𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
2928biimpd 217 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶𝑄𝑋 = 0 ))
3021, 29sylbid 228 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑄 𝑄) → 𝑋 = 0 ))
3117, 30syl5 33 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋𝐶(𝑃 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
3231expd 450 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋𝐶(𝑃 𝑄) → (𝑃 = 𝑄𝑋 = 0 )))
33323impia 1252 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑃 = 𝑄𝑋 = 0 ))
3414, 33impbid 200 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  0.cp0 16806  ccvr 33370  Atomscatm 33371  AtLatcal 33372  HLchlt 33458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459
This theorem is referenced by:  cvrat2  33536  atcvrneN  33537  atcvrj2b  33539
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