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Theorem isat3 35195
Description: The predicate "is an atom". (elat2 29590 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 2765 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 isat3.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 35174 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
6 simpl 474 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝐾 ∈ AtLat)
71, 2atl0cl 35191 . . . . . . 7 (𝐾 ∈ AtLat → 0𝐵)
87adantr 472 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 0𝐵)
9 simpr 477 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝑃𝐵)
10 isat3.l . . . . . . 7 = (le‘𝐾)
11 eqid 2765 . . . . . . 7 (lt‘𝐾) = (lt‘𝐾)
121, 10, 11, 3cvrval2 35162 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0𝐵𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
136, 8, 9, 12syl3anc 1490 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
141, 11, 2atlltn0 35194 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 (lt‘𝐾)𝑃𝑃0 ))
151, 11, 2atlltn0 35194 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1615adantlr 706 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1716imbi1d 332 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → (( 0 (lt‘𝐾)𝑥𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃)))
1817imbi2d 331 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃))))
19 impexp 441 . . . . . . . . 9 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)))
20 bi2.04 377 . . . . . . . . 9 (( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
2119, 20bitri 266 . . . . . . . 8 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
22 orcom 896 . . . . . . . . . 10 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥 = 0𝑥 = 𝑃))
23 neor 3028 . . . . . . . . . 10 ((𝑥 = 0𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃))
2422, 23bitri 266 . . . . . . . . 9 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥0𝑥 = 𝑃))
2524imbi2i 327 . . . . . . . 8 ((𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃)))
2618, 21, 253bitr4g 305 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2726ralbidva 3132 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → (∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2814, 27anbi12d 624 . . . . 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 271 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ((𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
3029pm5.32da 574 . . 3 (𝐾 ∈ AtLat → ((𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))) ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
315, 30bitr4d 273 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))))
32 3anass 1116 . 2 ((𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
3331, 32syl6bbr 280 1 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055   class class class wbr 4809  cfv 6068  Basecbs 16132  lecple 16223  ltcplt 17209  0.cp0 17305  ccvr 35150  Atomscatm 35151  AtLatcal 35152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-plt 17226  df-glb 17243  df-p0 17307  df-covers 35154  df-ats 35155  df-atl 35186
This theorem is referenced by:  atn0  35196  dihlspsnat  37221
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