Theorem 1cvratlt 34760

Description: An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)

Hypotheses

Ref	Expression
1cvratlt.b	|- B = ( Base ` K )
1cvratlt.l	|- .<_ = ( le ` K )
1cvratlt.s	|- .< = ( lt ` K )
1cvratlt.u	|- .1. = ( 1. ` K )
1cvratlt.c	|- C = ( <o ` K )
1cvratlt.a	|- A = ( Atoms ` K )

Assertion

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

Proof of Theorem 1cvratlt

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> K e. HL )
2		simpl3 1066	. . 3 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> X e. B )
3		simprl 794	. . 3 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> X C .1. )
4		1cvratlt.b	. . . 4 |- B = ( Base ` K )
5		1cvratlt.s	. . . 4 |- .< = ( lt ` K )
6		1cvratlt.u	. . . 4 |- .1. = ( 1. ` K )
7		1cvratlt.c	. . . 4 |- C = ( <o ` K )
8		1cvratlt.a	. . . 4 |- A = ( Atoms ` K )
9	4, 5, 6, 7, 8	1cvratex 34759	. . 3 |- ( ( K e. HL /\ X e. B /\ X C .1. ) -> E. q e. A q .< X )
10	1, 2, 3, 9	syl3anc 1326	. 2 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> E. q e. A q .< X )
11		simp1l1 1154	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> K e. HL )
12		simp1l2 1155	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> P e. A )
13		simp2 1062	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> q e. A )
14		simp1l3 1156	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> X e. B )
15		simp1rr 1127	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> P .<_ X )
16		simp3 1063	. . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> q .< X )
17		1cvratlt.l	. . . . 5 |- .<_ = ( le ` K )
18	4, 17, 5, 8	atlelt 34724	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ q e. A /\ X e. B ) /\ ( P .<_ X /\ q .< X ) ) -> P .< X )
19	11, 12, 13, 14, 15, 16, 18	syl132anc 1344	. . 3 |- ( ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) /\ q e. A /\ q .< X ) -> P .< X )
20	19	rexlimdv3a 3033	. 2 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> ( E. q e. A q .< X -> P .< X ) )
21	10, 20	mpd 15	1 |- ( ( ( K e. HL /\ P e. A /\ X e. B ) /\ ( X C .1. /\ P .<_ X ) ) -> P .< X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   1.cp1 17038   <o ccvr 34549   Atomscatm 34550   HLchlt 34637

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-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  cdlemb  35080  lhplt  35286