Theorem isat3 37248

Description: The predicate "is an atom". (elat2 30603 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 2738	. . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		isat3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	isat 37227	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
6		simpl 482	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝐾 ∈ AtLat)
7	1, 2	atl0cl 37244	. . . . . . 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 2738	. . . . . . 7 ⊢ (lt‘𝐾) = (lt‘𝐾)
12	1, 10, 11, 3	cvrval2 37215	. . . . . 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 1369	. . . . 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 37247	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑃 ↔ 𝑃 ≠ 0 ))
15	1, 11, 2	atlltn0 37247	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
16	15	adantlr 711	. . . . . . . . . 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		bi2.04 388	. . . . . . . . 9 ⊢ (( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
21	19, 20	bitri 274	. . . . . . . 8 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
22		orcom 866	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 = 0 ∨ 𝑥 = 𝑃))
23		neor 3035	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∨ 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
24	22, 23	bitri 274	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
25	24	imbi2i 335	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃)))
26	18, 21, 25	3bitr4g 313	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))
27	26	ralbidva 3119	. . . . . 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 630	. . . . 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 279	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ((𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
30	29	pm5.32da 578	. . 3 ⊢ (𝐾 ∈ AtLat → ((𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))) ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
31	5, 30	bitr4d 281	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))))
32		3anass 1093	. 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 288	1 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   class class class wbr 5070  ‘cfv 6418  Basecbs 16840  lecple 16895  ltcplt 17941  0.cp0 18056   ⋖ ccvr 37203  Atomscatm 37204  AtLatcal 37205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-plt 17963  df-glb 17980  df-p0 18058  df-covers 37207  df-ats 37208  df-atl 37239

This theorem is referenced by:  atn0  37249  dihlspsnat  39274