Theorem isat3 39680

Description: The predicate "is an atom". (elat2 32427 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 2737	. . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		isat3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	isat 39659	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
6		simpl 482	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝐾 ∈ AtLat)
7	1, 2	atl0cl 39676	. . . . . . 7 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵)
8	7	adantr 480	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 0 ∈ 𝐵)
9		simpr 484	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝑃 ∈ 𝐵)
10		isat3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11		eqid 2737	. . . . . . 7 ⊢ (lt‘𝐾) = (lt‘𝐾)
12	1, 10, 11, 3	cvrval2 39647	. . . . . 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 1374	. . . . 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 39679	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑃 ↔ 𝑃 ≠ 0 ))
15	1, 11, 2	atlltn0 39679	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
16	15	adantlr 716	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
17	16	imbi1d 341	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃)))
18	17	imbi2d 340	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃))))
19		impexp 450	. . . . . . . . 9 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)))
20		bi2.04 387	. . . . . . . . 9 ⊢ (( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
21	19, 20	bitri 275	. . . . . . . 8 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
22		orcom 871	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 = 0 ∨ 𝑥 = 𝑃))
23		neor 3025	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∨ 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
24	22, 23	bitri 275	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
25	24	imbi2i 336	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃)))
26	18, 21, 25	3bitr4g 314	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))
27	26	ralbidva 3159	. . . . . 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 633	. . . . 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 280	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ((𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
30	29	pm5.32da 579	. . 3 ⊢ (𝐾 ∈ AtLat → ((𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))) ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
31	5, 30	bitr4d 282	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))))
32		3anass 1095	. 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 289	1 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  Basecbs 17148  lecple 17196  ltcplt 18243  0.cp0 18356   ⋖ ccvr 39635  Atomscatm 39636  AtLatcal 39637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-plt 18263  df-glb 18280  df-p0 18358  df-covers 39639  df-ats 39640  df-atl 39671

This theorem is referenced by:  atn0  39681  dihlspsnat  41706