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Theorem lnnat 37936
Description: A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
lnnat.j ∨ = (joinβ€˜πΎ)
lnnat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
lnnat ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴))

Proof of Theorem lnnat
StepHypRef Expression
1 simpl1 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝐾 ∈ HL)
2 simpl2 1193 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 ∈ 𝐴)
3 eqid 2733 . . . . . . 7 (0.β€˜πΎ) = (0.β€˜πΎ)
4 eqid 2733 . . . . . . 7 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
5 lnnat.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
63, 4, 5atcvr0 37796 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) β†’ (0.β€˜πΎ)( β‹– β€˜πΎ)𝑃)
71, 2, 6syl2anc 585 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (0.β€˜πΎ)( β‹– β€˜πΎ)𝑃)
8 lnnat.j . . . . . . 7 ∨ = (joinβ€˜πΎ)
98, 4, 5atcvr1 37926 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ 𝑃( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)))
109biimpa 478 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑃( β‹– β€˜πΎ)(𝑃 ∨ 𝑄))
11 hlop 37870 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
12 eqid 2733 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1312, 3op0cl 37692 . . . . . . 7 (𝐾 ∈ OP β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
141, 11, 133syl 18 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
1512, 5atbase 37797 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
162, 15syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
171hllatd 37872 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝐾 ∈ Lat)
18 simpl3 1194 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑄 ∈ 𝐴)
1912, 5atbase 37797 . . . . . . . 8 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2018, 19syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2112, 8latjcl 18333 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2217, 16, 20, 21syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2312, 4cvrntr 37934 . . . . . 6 ((𝐾 ∈ HL ∧ ((0.β€˜πΎ) ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ (((0.β€˜πΎ)( β‹– β€˜πΎ)𝑃 ∧ 𝑃( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)) β†’ Β¬ (0.β€˜πΎ)( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)))
241, 14, 16, 22, 23syl13anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (((0.β€˜πΎ)( β‹– β€˜πΎ)𝑃 ∧ 𝑃( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)) β†’ Β¬ (0.β€˜πΎ)( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)))
257, 10, 24mp2and 698 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ Β¬ (0.β€˜πΎ)( β‹– β€˜πΎ)(𝑃 ∨ 𝑄))
26 simpll1 1213 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑄) ∈ 𝐴) β†’ 𝐾 ∈ HL)
273, 4, 5atcvr0 37796 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ 𝐴) β†’ (0.β€˜πΎ)( β‹– β€˜πΎ)(𝑃 ∨ 𝑄))
2826, 27sylancom 589 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ∨ 𝑄) ∈ 𝐴) β†’ (0.β€˜πΎ)( β‹– β€˜πΎ)(𝑃 ∨ 𝑄))
2925, 28mtand 815 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴)
3029ex 414 . 2 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 β†’ Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴))
318, 5hlatjidm 37877 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) β†’ (𝑃 ∨ 𝑃) = 𝑃)
32313adant3 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑃) = 𝑃)
33 simp2 1138 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
3432, 33eqeltrd 2834 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑃) ∈ 𝐴)
35 oveq2 7366 . . . . 5 (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑃) = (𝑃 ∨ 𝑄))
3635eleq1d 2819 . . . 4 (𝑃 = 𝑄 β†’ ((𝑃 ∨ 𝑃) ∈ 𝐴 ↔ (𝑃 ∨ 𝑄) ∈ 𝐴))
3734, 36syl5ibcom 244 . . 3 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑄) ∈ 𝐴))
3837necon3bd 2954 . 2 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴 β†’ 𝑃 β‰  𝑄))
3930, 38impbid 211 1 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  joincjn 18205  0.cp0 18317  Latclat 18325  OPcops 37680   β‹– ccvr 37770  Atomscatm 37771  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:  2atjlej  37988  cdleme11h  38775
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