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 → (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 2825	. . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		isat3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	isat 35354	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
6		simpl 476	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝐾 ∈ AtLat)
7	1, 2	atl0cl 35371	. . . . . . 7 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵)
8	7	adantr 474	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 0 ∈ 𝐵)
9		simpr 479	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → 𝑃 ∈ 𝐵)
10		isat3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
11		eqid 2825	. . . . . . 7 ⊢ (lt‘𝐾) = (lt‘𝐾)
12	1, 10, 11, 3	cvrval2 35342	. . . . . 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 1494	. . . . 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 35374	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑃 ↔ 𝑃 ≠ 0 ))
15	1, 11, 2	atlltn0 35374	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
16	15	adantlr 706	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑥 ↔ 𝑥 ≠ 0 ))
17	16	imbi1d 333	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃)))
18	17	imbi2d 332	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃))))
19		impexp 443	. . . . . . . . 9 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ ( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)))
20		bi2.04 379	. . . . . . . . 9 ⊢ (( 0 (lt‘𝐾)𝑥 → (𝑥 ≤ 𝑃 → 𝑥 = 𝑃)) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
21	19, 20	bitri 267	. . . . . . . 8 ⊢ ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → ( 0 (lt‘𝐾)𝑥 → 𝑥 = 𝑃)))
22		orcom 901	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 = 0 ∨ 𝑥 = 𝑃))
23		neor 3090	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∨ 𝑥 = 𝑃) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
24	22, 23	bitri 267	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ 𝑥 = 0 ) ↔ (𝑥 ≠ 0 → 𝑥 = 𝑃))
25	24	imbi2i 328	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )) ↔ (𝑥 ≤ 𝑃 → (𝑥 ≠ 0 → 𝑥 = 𝑃)))
26	18, 21, 25	3bitr4g 306	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((( 0 (lt‘𝐾)𝑥 ∧ 𝑥 ≤ 𝑃) → 𝑥 = 𝑃) ↔ (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))
27	26	ralbidva 3194	. . . . . 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 624	. . . . 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 272	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐵) → ((𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) ↔ 0 ( ⋖ ‘𝐾)𝑃))
30	29	pm5.32da 574	. . 3 ⊢ (𝐾 ∈ AtLat → ((𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))) ↔ (𝑃 ∈ 𝐵 ∧ 0 ( ⋖ ‘𝐾)𝑃)))
31	5, 30	bitr4d 274	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ (𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))))
32		3anass 1120	. 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	syl6bbr 281	1 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 )))))

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