Theorem riotaocN 34496

Description: The orthocomplement of the unique poset element such that ps. (riotaneg 11002 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
riotaoc.b	|- B = ( Base ` K )
riotaoc.o	|- ._|_ = ( oc ` K )
riotaoc.a	|- ( x = ( ._|_ ` y ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
riotaocN	|- ( ( K e. OP /\ E! x e. B ph ) -> ( iota_ x e. B ph ) = ( ._|_ ` ( iota_ y e. B ps ) ) )

Distinct variable groups:   x, y, B   x, K, y   x, ._|_ , y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem riotaocN

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ y ._|_
2		nfriota1 6618	. . 3 |- F/_ y ( iota_ y e. B ps )
3	1, 2	nffv 6198	. 2 |- F/_ y ( ._|_ ` ( iota_ y e. B ps ) )
4		riotaoc.b	. . 3 |- B = ( Base ` K )
5		riotaoc.o	. . 3 |- ._|_ = ( oc ` K )
6	4, 5	opoccl 34481	. 2 |- ( ( K e. OP /\ y e. B ) -> ( ._|_ ` y ) e. B )
7	4, 5	opoccl 34481	. 2 |- ( ( K e. OP /\ ( iota_ y e. B ps ) e. B ) -> ( ._|_ ` ( iota_ y e. B ps ) ) e. B )
8		riotaoc.a	. 2 |- ( x = ( ._|_ ` y ) -> ( ph <-> ps ) )
9		fveq2 6191	. 2 |- ( y = ( iota_ y e. B ps ) -> ( ._|_ ` y ) = ( ._|_ ` ( iota_ y e. B ps ) ) )
10	4, 5	opoccl 34481	. . 3 |- ( ( K e. OP /\ x e. B ) -> ( ._|_ ` x ) e. B )
11	4, 5	opcon2b 34484	. . 3 |- ( ( K e. OP /\ x e. B /\ y e. B ) -> ( x = ( ._|_ ` y ) <-> y = ( ._|_ ` x ) ) )
12	10, 11	reuhypd 4895	. 2 |- ( ( K e. OP /\ x e. B ) -> E! y e. B x = ( ._|_ ` y ) )
13	3, 6, 7, 8, 9, 12	riotaxfrd 6642	1 |- ( ( K e. OP /\ E! x e. B ph ) -> ( iota_ x e. B ph ) = ( ._|_ ` ( iota_ y e. B ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   ` cfv 5888   iota_crio 6610   Basecbs 15857   occoc 15949   OPcops 34459

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-nul 4789

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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-dm 5124  df-iota 5851  df-fv 5896  df-riota 6611  df-ov 6653  df-oposet 34463

This theorem is referenced by:  glbconN  34663