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Theorem isat3 39308
Description: The predicate "is an atom". (elat2 32359 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 2737 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 isat3.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 39287 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
6 simpl 482 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝐾 ∈ AtLat)
71, 2atl0cl 39304 . . . . . . 7 (𝐾 ∈ AtLat → 0𝐵)
87adantr 480 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 0𝐵)
9 simpr 484 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝑃𝐵)
10 isat3.l . . . . . . 7 = (le‘𝐾)
11 eqid 2737 . . . . . . 7 (lt‘𝐾) = (lt‘𝐾)
121, 10, 11, 3cvrval2 39275 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0𝐵𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
136, 8, 9, 12syl3anc 1373 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
141, 11, 2atlltn0 39307 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 (lt‘𝐾)𝑃𝑃0 ))
151, 11, 2atlltn0 39307 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1615adantlr 715 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1716imbi1d 341 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → (( 0 (lt‘𝐾)𝑥𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃)))
1817imbi2d 340 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃))))
19 impexp 450 . . . . . . . . 9 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)))
20 bi2.04 387 . . . . . . . . 9 (( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
2119, 20bitri 275 . . . . . . . 8 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
22 orcom 871 . . . . . . . . . 10 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥 = 0𝑥 = 𝑃))
23 neor 3034 . . . . . . . . . 10 ((𝑥 = 0𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃))
2422, 23bitri 275 . . . . . . . . 9 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥0𝑥 = 𝑃))
2524imbi2i 336 . . . . . . . 8 ((𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃)))
2618, 21, 253bitr4g 314 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2726ralbidva 3176 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → (∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2814, 27anbi12d 632 . . . . 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 280 . . . 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 282 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))))
32 3anass 1095 . 2 ((𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
3331, 32bitr4di 289 1 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  ltcplt 18354  0.cp0 18468  ccvr 39263  Atomscatm 39264  AtLatcal 39265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-plt 18375  df-glb 18392  df-p0 18470  df-covers 39267  df-ats 39268  df-atl 39299
This theorem is referenced by:  atn0  39309  dihlspsnat  41335
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