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Theorem isat3 35375
Description: The predicate "is an atom". (elat2 29743 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 2825 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 isat3.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 35354 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵0 ( ⋖ ‘𝐾)𝑃)))
6 simpl 476 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝐾 ∈ AtLat)
71, 2atl0cl 35371 . . . . . . 7 (𝐾 ∈ AtLat → 0𝐵)
87adantr 474 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 0𝐵)
9 simpr 479 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → 𝑃𝐵)
10 isat3.l . . . . . . 7 = (le‘𝐾)
11 eqid 2825 . . . . . . 7 (lt‘𝐾) = (lt‘𝐾)
121, 10, 11, 3cvrval2 35342 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0𝐵𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
136, 8, 9, 12syl3anc 1494 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥𝐵 (( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃))))
141, 11, 2atlltn0 35374 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐵) → ( 0 (lt‘𝐾)𝑃𝑃0 ))
151, 11, 2atlltn0 35374 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1615adantlr 706 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ( 0 (lt‘𝐾)𝑥𝑥0 ))
1716imbi1d 333 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → (( 0 (lt‘𝐾)𝑥𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃)))
1817imbi2d 332 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃))))
19 impexp 443 . . . . . . . . 9 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)))
20 bi2.04 379 . . . . . . . . 9 (( 0 (lt‘𝐾)𝑥 → (𝑥 𝑃𝑥 = 𝑃)) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
2119, 20bitri 267 . . . . . . . 8 ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → ( 0 (lt‘𝐾)𝑥𝑥 = 𝑃)))
22 orcom 901 . . . . . . . . . 10 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥 = 0𝑥 = 𝑃))
23 neor 3090 . . . . . . . . . 10 ((𝑥 = 0𝑥 = 𝑃) ↔ (𝑥0𝑥 = 𝑃))
2422, 23bitri 267 . . . . . . . . 9 ((𝑥 = 𝑃𝑥 = 0 ) ↔ (𝑥0𝑥 = 𝑃))
2524imbi2i 328 . . . . . . . 8 ((𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )) ↔ (𝑥 𝑃 → (𝑥0𝑥 = 𝑃)))
2618, 21, 253bitr4g 306 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐵) ∧ 𝑥𝐵) → ((( 0 (lt‘𝐾)𝑥𝑥 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))
2726ralbidva 3194 . . . . . 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 272 . . . 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 274 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))))))
32 3anass 1120 . 2 ((𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 ))) ↔ (𝑃𝐵 ∧ (𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
3331, 32syl6bbr 281 1 (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117   class class class wbr 4873  cfv 6123  Basecbs 16222  lecple 16312  ltcplt 17294  0.cp0 17390  ccvr 35330  Atomscatm 35331  AtLatcal 35332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-plt 17311  df-glb 17328  df-p0 17392  df-covers 35334  df-ats 35335  df-atl 35366
This theorem is referenced by:  atn0  35376  dihlspsnat  37401
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