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Theorem isat3 34095
Description: The predicate "is an atom". (elat2 29060 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 2621 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 isat3.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 34074 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
6 simpl 473 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝐾 ∈ AtLat)
71, 2atl0cl 34091 . . . . . . 7 (𝐾 ∈ AtLat → 0𝐵)
87adantr 481 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 0𝐵)
9 simpr 477 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝑃𝐵)
10 isat3.l . . . . . . 7 = (le‘𝐾)
11 eqid 2621 . . . . . . 7 (lt‘𝐾) = (lt‘𝐾)
121, 10, 11, 3cvrval2 34062 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0𝐵𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
136, 8, 9, 12syl3anc 1323 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
141, 11, 2atlltn0 34094 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 (lt‘𝐾)𝑃𝑃0 ))
151, 11, 2atlltn0 34094 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1615adantlr 750 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1716imbi1d 331 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → (( 0 (lt‘𝐾)𝑥𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃)))
1817imbi2d 330 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃))))
19 impexp 462 . . . . . . . . 9 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)))
20 bi2.04 376 . . . . . . . . 9 (( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
2119, 20bitri 264 . . . . . . . 8 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
22 orcom 402 . . . . . . . . . 10 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥 = 0𝑥 = 𝑃))
23 neor 2881 . . . . . . . . . 10 ((𝑥 = 0𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃))
2422, 23bitri 264 . . . . . . . . 9 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥0𝑥 = 𝑃))
2524imbi2i 326 . . . . . . . 8 ((𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃)))
2618, 21, 253bitr4g 303 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2726ralbidva 2979 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → (∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2814, 27anbi12d 746 . . . . 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 269 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ((𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
3029pm5.32da 672 . . 3 (𝐾 ∈ AtLat → ((𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))) ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
315, 30bitr4d 271 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))))
32 3anass 1040 . 2 ((𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
3331, 32syl6bbr 278 1 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907   class class class wbr 4615  cfv 5849  Basecbs 15784  lecple 15872  ltcplt 16865  0.cp0 16961  ccvr 34050  Atomscatm 34051  AtLatcal 34052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-plt 16882  df-glb 16899  df-p0 16963  df-covers 34054  df-ats 34055  df-atl 34086
This theorem is referenced by:  atn0  34096  dihlspsnat  36123
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