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Theorem atcvrj0 37937
Description: Two atoms covering the zero subspace are equal. (atcv1 31364 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 5109 . . . . . . . 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 483 . . . . . 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 37935 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
983adant3r1 1183 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
109adantr 482 . . . . . 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 414 . . . 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 1118 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = 0 β†’ 𝑃 = 𝑄))
15 oveq1 7365 . . . . . . 7 (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
1615breq2d 5118 . . . . . 6 (𝑃 = 𝑄 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) ↔ 𝑋𝐢(𝑄 ∨ 𝑄)))
1716biimpac 480 . . . . 5 ((𝑋𝐢(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) β†’ 𝑋𝐢(𝑄 ∨ 𝑄))
18 simpr3 1197 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
194, 7hlatjidm 37877 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) β†’ (𝑄 ∨ 𝑄) = 𝑄)
2018, 19syldan 592 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄 ∨ 𝑄) = 𝑄)
2120breq2d 5118 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑄 ∨ 𝑄) ↔ 𝑋𝐢𝑄))
22 hlatl 37868 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
2322adantr 482 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ AtLat)
24 simpr1 1195 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
25 atcvrj0.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
26 eqid 2733 . . . . . . . . 9 (leβ€˜πΎ) = (leβ€˜πΎ)
2725, 26, 5, 6, 7atcvreq0 37822 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋𝐢𝑄 ↔ 𝑋 = 0 ))
2823, 24, 18, 27syl3anc 1372 . . . . . . 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 417 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑃 = 𝑄 β†’ 𝑋 = 0 )))
33323impia 1118 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  0.cp0 18317   β‹– ccvr 37770  Atomscatm 37771  AtLatcal 37772  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  cvrat2  37938  atcvrneN  37939  atcvrj2b  37941
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