Theorem isat3 34594

Description: The predicate "is an atom". (elat2 29199 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 e. AtLat -> ( P e. A <-> ( P e. B /\ P =/= .0. /\ A. x e. 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 2622	. . . 4 |- ( <o ` K ) = ( <o ` K )
4		isat3.a	. . . 4 |- A = ( Atoms ` K )
5	1, 2, 3, 4	isat 34573	. . 3 |- ( K e. AtLat -> ( P e. A <-> ( P e. B /\ .0. ( <o ` K ) P ) ) )
6		simpl 473	. . . . . 6 |- ( ( K e. AtLat /\ P e. B ) -> K e. AtLat )
7	1, 2	atl0cl 34590	. . . . . . 7 |- ( K e. AtLat -> .0. e. B )
8	7	adantr 481	. . . . . 6 |- ( ( K e. AtLat /\ P e. B ) -> .0. e. B )
9		simpr 477	. . . . . 6 |- ( ( K e. AtLat /\ P e. B ) -> P e. B )
10		isat3.l	. . . . . . 7 |- .<_ = ( le ` K )
11		eqid 2622	. . . . . . 7 |- ( lt ` K ) = ( lt ` K )
12	1, 10, 11, 3	cvrval2 34561	. . . . . 6 |- ( ( K e. AtLat /\ .0. e. B /\ P e. B ) -> ( .0. ( <o ` K ) P <-> ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) ) )
13	6, 8, 9, 12	syl3anc 1326	. . . . 5 |- ( ( K e. AtLat /\ P e. B ) -> ( .0. ( <o ` K ) P <-> ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) ) )
14	1, 11, 2	atlltn0 34593	. . . . . 6 |- ( ( K e. AtLat /\ P e. B ) -> ( .0. ( lt ` K ) P <-> P =/= .0. ) )
15	1, 11, 2	atlltn0 34593	. . . . . . . . . . 11 |- ( ( K e. AtLat /\ x e. B ) -> ( .0. ( lt ` K ) x <-> x =/= .0. ) )
16	15	adantlr 751	. . . . . . . . . 10 |- ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( .0. ( lt ` K ) x <-> x =/= .0. ) )
17	16	imbi1d 331	. . . . . . . . 9 |- ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( .0. ( lt ` K ) x -> x = P ) <-> ( x =/= .0. -> x = P ) ) )
18	17	imbi2d 330	. . . . . . . 8 |- ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) <-> ( x .<_ P -> ( x =/= .0. -> x = P ) ) ) )
19		impexp 462	. . . . . . . . 9 |- ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( .0. ( lt ` K ) x -> ( x .<_ P -> x = P ) ) )
20		bi2.04 376	. . . . . . . . 9 |- ( ( .0. ( lt ` K ) x -> ( x .<_ P -> x = P ) ) <-> ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) )
21	19, 20	bitri 264	. . . . . . . 8 |- ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) )
22		orcom 402	. . . . . . . . . 10 |- ( ( x = P \/ x = .0. ) <-> ( x = .0. \/ x = P ) )
23		neor 2885	. . . . . . . . . 10 |- ( ( x = .0. \/ x = P ) <-> ( x =/= .0. -> x = P ) )
24	22, 23	bitri 264	. . . . . . . . 9 |- ( ( x = P \/ x = .0. ) <-> ( x =/= .0. -> x = P ) )
25	24	imbi2i 326	. . . . . . . 8 |- ( ( x .<_ P -> ( x = P \/ x = .0. ) ) <-> ( x .<_ P -> ( x =/= .0. -> x = P ) ) )
26	18, 21, 25	3bitr4g 303	. . . . . . 7 |- ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( x .<_ P -> ( x = P \/ x = .0. ) ) ) )
27	26	ralbidva 2985	. . . . . 6 |- ( ( K e. AtLat /\ P e. B ) -> ( A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) )
28	14, 27	anbi12d 747	. . . . 5 |- ( ( K e. AtLat /\ P e. B ) -> ( ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) <-> ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )
29	13, 28	bitr2d 269	. . . 4 |- ( ( K e. AtLat /\ P e. B ) -> ( ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) <-> .0. ( <o ` K ) P ) )
30	29	pm5.32da 673	. . 3 |- ( K e. AtLat -> ( ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) <-> ( P e. B /\ .0. ( <o ` K ) P ) ) )
31	5, 30	bitr4d 271	. 2 |- ( K e. AtLat -> ( P e. A <-> ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) ) )
32		3anass 1042	. 2 |- ( ( P e. B /\ P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) <-> ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )
33	31, 32	syl6bbr 278	1 |- ( K e. AtLat -> ( P e. A <-> ( P e. B /\ P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   0.cp0 17037   <o ccvr 34549   Atomscatm 34550   AtLatcal 34551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-plt 16958  df-glb 16975  df-p0 17039  df-covers 34553  df-ats 34554  df-atl 34585

This theorem is referenced by:  atn0  34595  dihlspsnat  36622