Theorem cvrval2 32834

Description: Binary relation expressing Y covers X. Definition of covers in [Kalmbach] p. 15. (cvbr2 27929 analog.) (Contributed by NM, 16-Nov-2011.)

Hypotheses

Ref	Expression
cvrletr.b	|- B = ( Base ` K )
cvrletr.l	|- .<_ = ( le ` K )
cvrletr.s	|- .< = ( lt ` K )
cvrletr.c	|- C = ( <o ` K )

Assertion

Ref	Expression
cvrval2	|- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) ) )

Distinct variable groups:   z, A   z, B   z, K   z, X   z, Y

Allowed substitution hints:   C( z)   .< ( z)   .<_ ( z)

Proof of Theorem cvrval2

Step	Hyp	Ref	Expression
1		cvrletr.b	. . 3 |- B = ( Base ` K )
2		cvrletr.s	. . 3 |- .< = ( lt ` K )
3		cvrletr.c	. . 3 |- C = ( <o ` K )
4	1, 2, 3	cvrval 32829	. 2 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
5		iman 426	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
6		df-ne 2623	. . . . . . . . 9 |- ( z =/= Y <-> -. z = Y )
7	6	anbi2i 699	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
8	5, 7	xchbinxr 313	. . . . . . 7 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) )
9		cvrletr.l	. . . . . . . . . . . . 13 |- .<_ = ( le ` K )
10	9, 2	pltval 16199	. . . . . . . . . . . 12 |- ( ( K e. A /\ z e. B /\ Y e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
11	10	3com23 1213	. . . . . . . . . . 11 |- ( ( K e. A /\ Y e. B /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
12	11	3expa 1207	. . . . . . . . . 10 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
13	12	anbi2d 709	. . . . . . . . 9 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( X .< z /\ z .< Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) ) )
14		anass 654	. . . . . . . . 9 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) )
15	13, 14	syl6rbbr 268	. . . . . . . 8 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ z .< Y ) ) )
16	15	notbid 296	. . . . . . 7 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> -. ( X .< z /\ z .< Y ) ) )
17	8, 16	syl5bb 261	. . . . . 6 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( X .< z /\ z .< Y ) ) )
18	17	ralbidva 2823	. . . . 5 |- ( ( K e. A /\ Y e. B ) -> ( A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> A. z e. B -. ( X .< z /\ z .< Y ) ) )
19		ralnex 2833	. . . . 5 |- ( A. z e. B -. ( X .< z /\ z .< Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) )
20	18, 19	syl6bb 265	. . . 4 |- ( ( K e. A /\ Y e. B ) -> ( A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) ) )
21	20	anbi2d 709	. . 3 |- ( ( K e. A /\ Y e. B ) -> ( ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
22	21	3adant2 1026	. 2 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
23	4, 22	bitr4d 260	1 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   =/= wne 2621   A.wral 2736   E.wrex 2737   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   ltcplt 16179   <o ccvr 32822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-plt 16197  df-covers 32826

This theorem is referenced by:  isat3  32867  cvlcvr1  32899