Theorem 1cvrat 33439

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 33327	. . . . . 6 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1009	. . . . 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 1021	. . . . . 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 33253	. . . . . 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 1022	. . . . . 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 33253	. . . . . 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 15343	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	2, 7, 10, 12	syl3anc 1219	. . . 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 6210	. . 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 15335	. . . . 5 |- ( ( K e. Lat /\ Q e. B /\ P e. B ) -> ( Q .\/ P ) e. B )
16	2, 10, 7, 15	syl3anc 1219	. . . 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 1023	. . . 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 15359	. . . 4 |- ( ( K e. Lat /\ ( Q .\/ P ) e. B /\ X e. B ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
20	2, 16, 17, 19	syl3anc 1219	. . 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 2493	. 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 988	. . 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 1168	. . 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 1024	. . . 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 2720	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q =/= P )
26		simp33 1026	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> -. P .<_ X )
27		hlop 33326	. . . . . 6 |- ( K e. HL -> K e. OP )
28	27	3ad2ant1 1009	. . . . 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 33155	. . . . 5 |- ( ( K e. OP /\ Q e. B ) -> Q .<_ .1. )
32	28, 10, 31	syl2anc 661	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ .1. )
33		simp32 1025	. . . . 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 33438	. . . . 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 1227	. . . 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 33405	. . . 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 1226	. 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 2540	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 965   = wceq 1370   e. wcel 1758   =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   1.cp1 15322   Latclat 15329   OPcops 33136   <o ccvr 33226   Atomscatm 33227   HLchlt 33314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315

This theorem is referenced by:  cdlemblem  33756  cdlemb  33757  lhpat  34006