Theorem isat3 40332

Description: The predicate "is an atom". (elat2 32924 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 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑃   𝑥, 0

Allowed substitution hints:   𝐴(𝑥)   ≤ (𝑥)

Proof of Theorem isat3

Step	Hyp	Ref	Expression
1		isat3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		isat3.z	. . . 4 ⊢ 0 = (0.‘𝐾)
3		eqid 2761	. . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		isat3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	isat 40311	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
6		simpl 488	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝐾 ∈ AtLat)
7	1, 2	atl0cl 40328	. . . . . . 7 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵)
8	7	adantr 486	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 0 ∈ 𝐵)
9		simpr 490	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝑃 ∈ 𝐵)
10		isat3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11		eqid 2761	. . . . . . 7 ⊢ (lt‘𝐾) = (lt‘𝐾)
12	1, 10, 11, 3	cvrval2 40299	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥 ∈ 𝐵 (( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃))))
13	6, 8, 9, 12	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ( 0 ( ⋖ ‘𝐾)𝑃 ↔ ( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥 ∈ 𝐵 (( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃))))
14	1, 11, 2	atlltn0 40331	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑃 ↔ 𝑃 ≠ 0 ))
15	1, 11, 2	atlltn0 40331	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
16	15	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
17	16	imbi1d 344	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃)))
18	17	imbi2d 343	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃))))
19		impexp 456	. . . . . . . . 9 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)))
20		bi2.04 392	. . . . . . . . 9 ⊢ (( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
21	19, 20	bitri 278	. . . . . . . 8 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
22		orcom 884	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 = 0 ∨ 𝑥 = 𝑃))
23		neor 3048	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∨ 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
24	22, 23	bitri 278	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
25	24	imbi2i 339	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃)))
26	18, 21, 25	3bitr4g 317	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))
27	26	ralbidva 3184	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))
28	14, 27	anbi12d 644	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → (( 0 (lt‘𝐾)𝑃 ∧ ∀𝑥 ∈ 𝐵 (( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃)) ↔ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))
29	13, 28	bitr2d 283	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ((𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
30	29	pm5.32da 590	. . 3 ⊢ (𝐾 ∈ AtLat → ((𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))) ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
31	5, 30	bitr4d 285	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))))
32		3anass 1111	. 2 ⊢ ((𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) ↔ (𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))
33	31, 32	bitr4di 292	1 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   class class class wbr 5103  ‘cfv 6531  Basecbs 17367  lecple 17415  ltcplt 18462  0.cp0 18575   ⋖ ccvr 40287  Atomscatm 40288  AtLatcal 40289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-plt 18482  df-glb 18499  df-p0 18577  df-covers 40291  df-ats 40292  df-atl 40323

This theorem is used by:  atn0  40333  dihlspsnat  42358