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Theorem isat3 36323
Description: The predicate "is an atom". (elat2 30044 analog.) (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
isat3.b 𝐵 = (Base‘𝐾)
isat3.l = (le‘𝐾)
isat3.z 0 = (0.‘𝐾)
isat3.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
isat3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑃   𝑥, 0
Allowed substitution hints:   𝐴(𝑥)   (𝑥)

Proof of Theorem isat3
StepHypRef Expression
1 isat3.b . . . 4 𝐵 = (Base‘𝐾)
2 isat3.z . . . 4 0 = (0.‘𝐾)
3 eqid 2818 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 isat3.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 36302 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
6 simpl 483 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝐾 ∈ AtLat)
71, 2atl0cl 36319 . . . . . . 7 (𝐾 ∈ AtLat → 0𝐵)
87adantr 481 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 0𝐵)
9 simpr 485 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝑃𝐵)
10 isat3.l . . . . . . 7 = (le‘𝐾)
11 eqid 2818 . . . . . . 7 (lt‘𝐾) = (lt‘𝐾)
121, 10, 11, 3cvrval2 36290 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0𝐵𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
136, 8, 9, 12syl3anc 1363 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
141, 11, 2atlltn0 36322 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 (lt‘𝐾)𝑃𝑃0 ))
151, 11, 2atlltn0 36322 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1615adantlr 711 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1716imbi1d 343 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → (( 0 (lt‘𝐾)𝑥𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃)))
1817imbi2d 342 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃))))
19 impexp 451 . . . . . . . . 9 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)))
20 bi2.04 389 . . . . . . . . 9 (( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
2119, 20bitri 276 . . . . . . . 8 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
22 orcom 864 . . . . . . . . . 10 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥 = 0𝑥 = 𝑃))
23 neor 3105 . . . . . . . . . 10 ((𝑥 = 0𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃))
2422, 23bitri 276 . . . . . . . . 9 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥0𝑥 = 𝑃))
2524imbi2i 337 . . . . . . . 8 ((𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃)))
2618, 21, 253bitr4g 315 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2726ralbidva 3193 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → (∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2814, 27anbi12d 630 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → (( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃)) ↔ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
2913, 28bitr2d 281 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ((𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
3029pm5.32da 579 . . 3 (𝐾 ∈ AtLat → ((𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))) ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
315, 30bitr4d 283 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))))
32 3anass 1087 . 2 ((𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
3331, 32syl6bbr 290 1 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  ltcplt 17539  0.cp0 17635  ccvr 36278  Atomscatm 36279  AtLatcal 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-plt 17556  df-glb 17573  df-p0 17637  df-covers 36282  df-ats 36283  df-atl 36314
This theorem is referenced by:  atn0  36324  dihlspsnat  38349
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