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Theorem cvrat2 38906
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 32215 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 2727 . . . . . . . . 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 38905 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
763expa 1115 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = (0.β€˜πΎ) ↔ 𝑃 = 𝑄))
87necon3bid 2981 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) ↔ 𝑃 β‰  𝑄))
9 simpl 481 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1191 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
11 hllat 38839 . . . . . . . . . . 11 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1211adantr 479 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
13 simpr2 1192 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
141, 5atbase 38765 . . . . . . . . . . 11 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1513, 14syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
16 simpr3 1193 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
171, 5atbase 38765 . . . . . . . . . . 11 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1816, 17syl 17 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
191, 2latjcl 18436 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2012, 15, 18, 19syl3anc 1368 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
21 eqid 2727 . . . . . . . . . . 11 (ltβ€˜πΎ) = (ltβ€˜πΎ)
221, 21, 4cvrlt 38746 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄))
2322ex 411 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋𝐢(𝑃 ∨ 𝑄) β†’ 𝑋(ltβ€˜πΎ)(𝑃 ∨ 𝑄)))
249, 10, 20, 23syl3anc 1368 . . . . . . . 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 38899 . . . . . . . . 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 415 . . . . . . . 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 405 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 β‰  (0.β€˜πΎ) β†’ 𝑋 ∈ 𝐴))
298, 28sylbird 259 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑃 β‰  𝑄 β†’ 𝑋 ∈ 𝐴))
3029ex 411 . . . 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 409 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
33323impia 1114 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936   class class class wbr 5150  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  ltcplt 18305  joincjn 18308  0.cp0 18420  Latclat 18428   β‹– ccvr 38738  Atomscatm 38739  HLchlt 38826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18292  df-poset 18310  df-plt 18327  df-lub 18343  df-glb 18344  df-join 18345  df-meet 18346  df-p0 18422  df-lat 18429  df-clat 18496  df-oposet 38652  df-ol 38654  df-oml 38655  df-covers 38742  df-ats 38743  df-atl 38774  df-cvlat 38798  df-hlat 38827
This theorem is referenced by:  cvrat3  38919  atcvrlln  38997  lncvrelatN  39258
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