Theorem 1cvrat 29958

Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)

Hypotheses

Ref	Expression
1cvrat.b	|- B = ( Base ` K )
1cvrat.l	|- .<_ = ( le ` K )
1cvrat.j	|- .\/ = ( join ` K )
1cvrat.m	|- ./\ = ( meet ` K )
1cvrat.u	|- .1. = ( 1. ` K )
1cvrat.c	|- C = ( <o ` K )
1cvrat.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
1cvrat	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) e. A )

Proof of Theorem 1cvrat

Step	Hyp	Ref	Expression
1		hllat 29846	. . . . . 6 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 978	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. Lat )
3		simp21 990	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P e. A )
4		1cvrat.b	. . . . . . 7 |- B = ( Base ` K )
5		1cvrat.a	. . . . . . 7 |- A = ( Atoms ` K )
6	4, 5	atbase 29772	. . . . . 6 |- ( P e. A -> P e. B )
7	3, 6	syl 16	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P e. B )
8		simp22 991	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q e. A )
9	4, 5	atbase 29772	. . . . . 6 |- ( Q e. A -> Q e. B )
10	8, 9	syl 16	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q e. B )
11		1cvrat.j	. . . . . 6 |- .\/ = ( join ` K )
12	4, 11	latjcom 14443	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	2, 7, 10, 12	syl3anc 1184	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
14	13	oveq1d 6055	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = ( ( Q .\/ P ) ./\ X ) )
15	4, 11	latjcl 14434	. . . . 5 |- ( ( K e. Lat /\ Q e. B /\ P e. B ) -> ( Q .\/ P ) e. B )
16	2, 10, 7, 15	syl3anc 1184	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( Q .\/ P ) e. B )
17		simp23 992	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> X e. B )
18		1cvrat.m	. . . . 5 |- ./\ = ( meet ` K )
19	4, 18	latmcom 14459	. . . 4 |- ( ( K e. Lat /\ ( Q .\/ P ) e. B /\ X e. B ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
20	2, 16, 17, 19	syl3anc 1184	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
21	14, 20	eqtrd 2436	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
22		simp1 957	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. HL )
23	17, 8, 3	3jca 1134	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X e. B /\ Q e. A /\ P e. A ) )
24		simp31 993	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P =/= Q )
25	24	necomd 2650	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q =/= P )
26		simp33 995	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> -. P .<_ X )
27		hlop 29845	. . . . . 6 |- ( K e. HL -> K e. OP )
28	27	3ad2ant1 978	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. OP )
29		1cvrat.l	. . . . . 6 |- .<_ = ( le ` K )
30		1cvrat.u	. . . . . 6 |- .1. = ( 1. ` K )
31	4, 29, 30	ople1 29674	. . . . 5 |- ( ( K e. OP /\ Q e. B ) -> Q .<_ .1. )
32	28, 10, 31	syl2anc 643	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ .1. )
33		simp32 994	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> X C .1. )
34		1cvrat.c	. . . . . 6 |- C = ( <o ` K )
35	4, 29, 11, 30, 34, 5	1cvrjat 29957	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( X C .1. /\ -. P .<_ X ) ) -> ( X .\/ P ) = .1. )
36	22, 17, 3, 33, 26, 35	syl32anc 1192	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X .\/ P ) = .1. )
37	32, 36	breqtrrd 4198	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ ( X .\/ P ) )
38	4, 29, 11, 18, 5	cvrat3 29924	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) -> ( ( Q =/= P /\ -. P .<_ X /\ Q .<_ ( X .\/ P ) ) -> ( X ./\ ( Q .\/ P ) ) e. A ) )
39	38	imp 419	. . 3 |- ( ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) /\ ( Q =/= P /\ -. P .<_ X /\ Q .<_ ( X .\/ P ) ) ) -> ( X ./\ ( Q .\/ P ) ) e. A )
40	22, 23, 25, 26, 37, 39	syl23anc 1191	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X ./\ ( Q .\/ P ) ) e. A )
41	21, 40	eqeltrd 2478	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   1.cp1 14422   Latclat 14429   OPcops 29655   <o ccvr 29745   Atomscatm 29746   HLchlt 29833

This theorem is referenced by:  cdlemblem  30275  cdlemb  30276  lhpat  30525

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834