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Theorem cvrat2 37938
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 31371 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat2.b 𝐡 = (Baseβ€˜πΎ)
cvrat2.j ∨ = (joinβ€˜πΎ)
cvrat2.c 𝐢 = ( β‹– β€˜πΎ)
cvrat2.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvrat2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)

Proof of Theorem cvrat2
StepHypRef Expression
1 cvrat2.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
2 cvrat2.j . . . . . . . . 9 ∨ = (joinβ€˜πΎ)
3 eqid 2733 . . . . . . . . 9 (0.β€˜πΎ) = (0.β€˜πΎ)
4 cvrat2.c . . . . . . . . 9 𝐢 = ( β‹– β€˜πΎ)
5 cvrat2.a . . . . . . . . 9 𝐴 = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5atcvrj0 37937 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
763expa 1119 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
87necon3bid 2985 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) ↔ 𝑃 β‰  𝑄))
9 simpl 484 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1195 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
11 hllat 37871 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1211adantr 482 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
13 simpr2 1196 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
141, 5atbase 37797 . . . . . . . . . . 11 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1513, 14syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
16 simpr3 1197 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
171, 5atbase 37797 . . . . . . . . . . 11 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1816, 17syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
191, 2latjcl 18333 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2012, 15, 18, 19syl3anc 1372 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
21 eqid 2733 . . . . . . . . . . 11 (ltβ€˜πΎ) = (ltβ€˜πΎ)
221, 21, 4cvrlt 37778 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄))
2322ex 414 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
249, 10, 20, 23syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
251, 21, 2, 3, 5cvrat 37931 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  (0.β€˜πΎ) ∧ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
2625expcomd 418 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴)))
2724, 26syld 47 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴)))
2827imp 408 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴))
298, 28sylbird 260 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴))
3029ex 414 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴)))
3130com23 86 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 β‰  𝑄 β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋 ∈ 𝐴)))
3231impd 412 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
33323impia 1118 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   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  ltcplt 18202  joincjn 18205  0.cp0 18317  Latclat 18325   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:  cvrat3  37951  atcvrlln  38029  lncvrelatN  38290
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