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Theorem isat3 38779
Description: The predicate "is an atom". (elat2 32149 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 2728 . . . 4 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
4 isat3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4isat 38758 . . 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 38775 . . . . . . 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 2728 . . . . . . 7 (ltβ€˜πΎ) = (ltβ€˜πΎ)
121, 10, 11, 3cvrval2 38746 . . . . . 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 1369 . . . . 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 38778 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)𝑃 ↔ 𝑃 β‰  0 ))
151, 11, 2atlltn0 38778 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ π‘₯ ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)π‘₯ ↔ π‘₯ β‰  0 ))
1615adantlr 714 . . . . . . . . . 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 869 . . . . . . . . . 10 ((π‘₯ = 𝑃 ∨ π‘₯ = 0 ) ↔ (π‘₯ = 0 ∨ π‘₯ = 𝑃))
23 neor 3031 . . . . . . . . . 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 3172 . . . . . 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 631 . . . . 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 1093 . 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 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058   class class class wbr 5148  β€˜cfv 6548  Basecbs 17179  lecple 17239  ltcplt 18299  0.cp0 18414   β‹– ccvr 38734  Atomscatm 38735  AtLatcal 38736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-plt 18321  df-glb 18338  df-p0 18416  df-covers 38738  df-ats 38739  df-atl 38770
This theorem is referenced by:  atn0  38780  dihlspsnat  40806
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