Theorem plttr 16970

Description: The less-than relation is transitive. (psstr 3711 analog.) (Contributed by NM, 2-Dec-2011.)

Hypotheses

Ref	Expression
pltnlt.b	|- B = ( Base ` K )
pltnlt.s	|- .< = ( lt ` K )

Assertion

Ref	Expression
plttr	|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> X .< Z ) )

Proof of Theorem plttr

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
2		pltnlt.s	. . . . . 6 |- .< = ( lt ` K )
3	1, 2	pltle 16961	. . . . 5 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> X ( le ` K ) Y ) )
4	3	3adant3r3 1276	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .< Y -> X ( le ` K ) Y ) )
5	1, 2	pltle 16961	. . . . 5 |- ( ( K e. Poset /\ Y e. B /\ Z e. B ) -> ( Y .< Z -> Y ( le ` K ) Z ) )
6	5	3adant3r1 1274	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .< Z -> Y ( le ` K ) Z ) )
7		pltnlt.b	. . . . 5 |- B = ( Base ` K )
8	7, 1	postr 16953	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) Z ) -> X ( le ` K ) Z ) )
9	4, 6, 8	syl2and 500	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> X ( le ` K ) Z ) )
10	7, 2	pltn2lp 16969	. . . . . 6 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> -. ( X .< Y /\ Y .< X ) )
11	10	3adant3r3 1276	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> -. ( X .< Y /\ Y .< X ) )
12		breq2 4657	. . . . . . 7 |- ( X = Z -> ( Y .< X <-> Y .< Z ) )
13	12	anbi2d 740	. . . . . 6 |- ( X = Z -> ( ( X .< Y /\ Y .< X ) <-> ( X .< Y /\ Y .< Z ) ) )
14	13	notbid 308	. . . . 5 |- ( X = Z -> ( -. ( X .< Y /\ Y .< X ) <-> -. ( X .< Y /\ Y .< Z ) ) )
15	11, 14	syl5ibcom 235	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X = Z -> -. ( X .< Y /\ Y .< Z ) ) )
16	15	necon2ad 2809	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> X =/= Z ) )
17	9, 16	jcad 555	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> ( X ( le ` K ) Z /\ X =/= Z ) ) )
18	1, 2	pltval 16960	. . 3 |- ( ( K e. Poset /\ X e. B /\ Z e. B ) -> ( X .< Z <-> ( X ( le ` K ) Z /\ X =/= Z ) ) )
19	18	3adant3r2 1275	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .< Z <-> ( X ( le ` K ) Z /\ X =/= Z ) ) )
20	17, 19	sylibrd 249	1 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> X .< Z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   ltcplt 16941

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-preset 16928  df-poset 16946  df-plt 16958

This theorem is referenced by:  pltletr  16971  plelttr  16972  pospo  16973  archiabllem2c  29749  ofldchr  29814  hlhgt2  34675  hl0lt1N  34676  lhp0lt  35289