Theorem hlrelat 34688

Description: A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 29223 analog.) (Contributed by NM, 4-Feb-2012.)

Hypotheses

Ref	Expression
hlrelat5.b	|- B = ( Base ` K )
hlrelat5.l	|- .<_ = ( le ` K )
hlrelat5.s	|- .< = ( lt ` K )
hlrelat5.j	|- .\/ = ( join ` K )
hlrelat5.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
hlrelat	|- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) -> E. p e. A ( X .< ( X .\/ p ) /\ ( X .\/ p ) .<_ Y ) )

Distinct variable groups:   A, p   B, p   K, p   .<_ , p   X, p   Y, p   .< , p

Allowed substitution hint:   .\/ ( p)

Proof of Theorem hlrelat

Step	Hyp	Ref	Expression
1		hlrelat5.b	. . . 4 |- B = ( Base ` K )
2		hlrelat5.l	. . . 4 |- .<_ = ( le ` K )
3		hlrelat5.s	. . . 4 |- .< = ( lt ` K )
4		hlrelat5.a	. . . 4 |- A = ( Atoms ` K )
5	1, 2, 3, 4	hlrelat1 34686	. . 3 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X .< Y -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) ) )
6	5	imp 445	. 2 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) )
7		simpll1 1100	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> K e. HL )
8		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
9	7, 8	syl 17	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> K e. Lat )
10		simpll2 1101	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> X e. B )
11	1, 4	atbase 34576	. . . . . 6 |- ( p e. A -> p e. B )
12	11	adantl 482	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> p e. B )
13		hlrelat5.j	. . . . . 6 |- .\/ = ( join ` K )
14	1, 2, 3, 13	latnle 17085	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ p e. B ) -> ( -. p .<_ X <-> X .< ( X .\/ p ) ) )
15	9, 10, 12, 14	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> ( -. p .<_ X <-> X .< ( X .\/ p ) ) )
16	2, 3	pltle 16961	. . . . . . . 8 |- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X .< Y -> X .<_ Y ) )
17	16	imp 445	. . . . . . 7 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) -> X .<_ Y )
18	17	adantr 481	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> X .<_ Y )
19	18	biantrurd 529	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> ( p .<_ Y <-> ( X .<_ Y /\ p .<_ Y ) ) )
20		simpll3 1102	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> Y e. B )
21	1, 2, 13	latjle12 17062	. . . . . 6 |- ( ( K e. Lat /\ ( X e. B /\ p e. B /\ Y e. B ) ) -> ( ( X .<_ Y /\ p .<_ Y ) <-> ( X .\/ p ) .<_ Y ) )
22	9, 10, 12, 20, 21	syl13anc 1328	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> ( ( X .<_ Y /\ p .<_ Y ) <-> ( X .\/ p ) .<_ Y ) )
23	19, 22	bitrd 268	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> ( p .<_ Y <-> ( X .\/ p ) .<_ Y ) )
24	15, 23	anbi12d 747	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) /\ p e. A ) -> ( ( -. p .<_ X /\ p .<_ Y ) <-> ( X .< ( X .\/ p ) /\ ( X .\/ p ) .<_ Y ) ) )
25	24	rexbidva 3049	. 2 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) -> ( E. p e. A ( -. p .<_ X /\ p .<_ Y ) <-> E. p e. A ( X .< ( X .\/ p ) /\ ( X .\/ p ) .<_ Y ) ) )
26	6, 25	mpbid 222	1 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X .< Y ) -> E. p e. A ( X .< ( X .\/ p ) /\ ( X .\/ p ) .<_ Y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   ltcplt 16941   joincjn 16944   Latclat 17045   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-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:  hlrelat2  34689  atle  34722  2atlt  34725