Theorem 1cvrat 32842

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 32730	. . . . . 6 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1004	. . . . 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 1016	. . . . . 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 32656	. . . . . 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 1017	. . . . . 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 32656	. . . . . 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 15225	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	2, 7, 10, 12	syl3anc 1213	. . . 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 6105	. . 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 15217	. . . . 5 |- ( ( K e. Lat /\ Q e. B /\ P e. B ) -> ( Q .\/ P ) e. B )
16	2, 10, 7, 15	syl3anc 1213	. . . 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 1018	. . . 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 15241	. . . 4 |- ( ( K e. Lat /\ ( Q .\/ P ) e. B /\ X e. B ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
20	2, 16, 17, 19	syl3anc 1213	. . 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 2473	. 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 983	. . 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 1163	. . 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 1019	. . . 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 2693	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q =/= P )
26		simp33 1021	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> -. P .<_ X )
27		hlop 32729	. . . . . 6 |- ( K e. HL -> K e. OP )
28	27	3ad2ant1 1004	. . . . 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 32558	. . . . 5 |- ( ( K e. OP /\ Q e. B ) -> Q .<_ .1. )
32	28, 10, 31	syl2anc 656	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ .1. )
33		simp32 1020	. . . . 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 32841	. . . . 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 1221	. . . 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 4315	. . 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 32808	. . . 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 429	. . 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 1220	. 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 2515	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 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   1.cp1 15204   Latclat 15211   OPcops 32539   <o ccvr 32629   Atomscatm 32630   HLchlt 32717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718

This theorem is referenced by:  cdlemblem  33159  cdlemb  33160  lhpat  33409