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Theorem isat3 38680
Description: The predicate "is an atom". (elat2 32087 analog.) (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
isat3.b 𝐡 = (Baseβ€˜πΎ)
isat3.l ≀ = (leβ€˜πΎ)
isat3.z 0 = (0.β€˜πΎ)
isat3.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
isat3 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐾   π‘₯,𝑃   π‘₯, 0
Allowed substitution hints:   𝐴(π‘₯)   ≀ (π‘₯)

Proof of Theorem isat3
StepHypRef Expression
1 isat3.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 isat3.z . . . 4 0 = (0.β€˜πΎ)
3 eqid 2724 . . . 4 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
4 isat3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4isat 38659 . . 3 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 ( β‹– β€˜πΎ)𝑃)))
6 simpl 482 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 𝐾 ∈ AtLat)
71, 2atl0cl 38676 . . . . . . 7 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
87adantr 480 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 0 ∈ 𝐡)
9 simpr 484 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 𝑃 ∈ 𝐡)
10 isat3.l . . . . . . 7 ≀ = (leβ€˜πΎ)
11 eqid 2724 . . . . . . 7 (ltβ€˜πΎ) = (ltβ€˜πΎ)
121, 10, 11, 3cvrval2 38647 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡) β†’ ( 0 ( β‹– β€˜πΎ)𝑃 ↔ ( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃))))
136, 8, 9, 12syl3anc 1368 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ( 0 ( β‹– β€˜πΎ)𝑃 ↔ ( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃))))
141, 11, 2atlltn0 38679 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)𝑃 ↔ 𝑃 β‰  0 ))
151, 11, 2atlltn0 38679 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ π‘₯ ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)π‘₯ ↔ π‘₯ β‰  0 ))
1615adantlr 712 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)π‘₯ ↔ π‘₯ β‰  0 ))
1716imbi1d 341 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ (( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃)))
1817imbi2d 340 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ((π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))))
19 impexp 450 . . . . . . . . 9 ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ ( 0 (ltβ€˜πΎ)π‘₯ β†’ (π‘₯ ≀ 𝑃 β†’ π‘₯ = 𝑃)))
20 bi2.04 387 . . . . . . . . 9 (( 0 (ltβ€˜πΎ)π‘₯ β†’ (π‘₯ ≀ 𝑃 β†’ π‘₯ = 𝑃)) ↔ (π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)))
2119, 20bitri 275 . . . . . . . 8 ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ (π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)))
22 orcom 867 . . . . . . . . . 10 ((π‘₯ = 𝑃 ∨ π‘₯ = 0 ) ↔ (π‘₯ = 0 ∨ π‘₯ = 𝑃))
23 neor 3026 . . . . . . . . . 10 ((π‘₯ = 0 ∨ π‘₯ = 𝑃) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))
2422, 23bitri 275 . . . . . . . . 9 ((π‘₯ = 𝑃 ∨ π‘₯ = 0 ) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))
2524imbi2i 336 . . . . . . . 8 ((π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃)))
2618, 21, 253bitr4g 314 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))
2726ralbidva 3167 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))
2814, 27anbi12d 630 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ (( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃)) ↔ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
2913, 28bitr2d 280 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ((𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))) ↔ 0 ( β‹– β€˜πΎ)𝑃))
3029pm5.32da 578 . . 3 (𝐾 ∈ AtLat β†’ ((𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))) ↔ (𝑃 ∈ 𝐡 ∧ 0 ( β‹– β€˜πΎ)𝑃)))
315, 30bitr4d 282 . 2 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))))
32 3anass 1092 . 2 ((𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))) ↔ (𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
3331, 32bitr4di 289 1 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053   class class class wbr 5139  β€˜cfv 6534  Basecbs 17149  lecple 17209  ltcplt 18269  0.cp0 18384   β‹– ccvr 38635  Atomscatm 38636  AtLatcal 38637
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-plt 18291  df-glb 18308  df-p0 18386  df-covers 38639  df-ats 38640  df-atl 38671
This theorem is referenced by:  atn0  38681  dihlspsnat  40707
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