Theorem 1cvrat 32493

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 32381	. . . . . 6 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1018	. . . . 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 1030	. . . . . 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 32307	. . . . . 6 |- ( P e. A -> P e. B )
7	3, 6	syl 17	. . . . 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 1031	. . . . . 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 32307	. . . . . 6 |- ( Q e. A -> Q e. B )
10	8, 9	syl 17	. . . . 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 16013	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	2, 7, 10, 12	syl3anc 1230	. . . 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 6293	. . 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 16005	. . . . 5 |- ( ( K e. Lat /\ Q e. B /\ P e. B ) -> ( Q .\/ P ) e. B )
16	2, 10, 7, 15	syl3anc 1230	. . . 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 1032	. . . 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 16029	. . . 4 |- ( ( K e. Lat /\ ( Q .\/ P ) e. B /\ X e. B ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
20	2, 16, 17, 19	syl3anc 1230	. . 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 2443	. 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 997	. . 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 1177	. . 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 1033	. . . 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 2674	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q =/= P )
26		simp33 1035	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> -. P .<_ X )
27		hlop 32380	. . . . . 6 |- ( K e. HL -> K e. OP )
28	27	3ad2ant1 1018	. . . . 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 32209	. . . . 5 |- ( ( K e. OP /\ Q e. B ) -> Q .<_ .1. )
32	28, 10, 31	syl2anc 659	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ .1. )
33		simp32 1034	. . . . 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 32492	. . . . 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 1238	. . . 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 4421	. . 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 32459	. . . 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 427	. . 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 1237	. 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 2490	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   1.cp1 15992   Latclat 15999   OPcops 32190   <o ccvr 32280   Atomscatm 32281   HLchlt 32368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369

This theorem is referenced by:  cdlemblem  32810  cdlemb  32811  lhpat  33060