Theorem isst 29072

Description: Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
isst	|- ( S e. States <-> ( S : CH --> ( 0 [,] 1 ) /\ ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )

Distinct variable group:   x, y, S

Proof of Theorem isst

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( 0 [,] 1 ) e. _V
2		chex 28083	. . . 4 |- CH e. _V
3	1, 2	elmap 7886	. . 3 |- ( S e. ( ( 0 [,] 1 ) ^m CH ) <-> S : CH --> ( 0 [,] 1 ) )
4	3	anbi1i 731	. 2 |- ( ( S e. ( ( 0 [,] 1 ) ^m CH ) /\ ( ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) ) <-> ( S : CH --> ( 0 [,] 1 ) /\ ( ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) ) )
5		fveq1 6190	. . . . 5 |- ( f = S -> ( f ` ~H ) = ( S ` ~H ) )
6	5	eqeq1d 2624	. . . 4 |- ( f = S -> ( ( f ` ~H ) = 1 <-> ( S ` ~H ) = 1 ) )
7		fveq1 6190	. . . . . . 7 |- ( f = S -> ( f ` ( x vH y ) ) = ( S ` ( x vH y ) ) )
8		fveq1 6190	. . . . . . . 8 |- ( f = S -> ( f ` x ) = ( S ` x ) )
9		fveq1 6190	. . . . . . . 8 |- ( f = S -> ( f ` y ) = ( S ` y ) )
10	8, 9	oveq12d 6668	. . . . . . 7 |- ( f = S -> ( ( f ` x ) + ( f ` y ) ) = ( ( S ` x ) + ( S ` y ) ) )
11	7, 10	eqeq12d 2637	. . . . . 6 |- ( f = S -> ( ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) <-> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) )
12	11	imbi2d 330	. . . . 5 |- ( f = S -> ( ( x C_ ( _|_ ` y ) -> ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) ) <-> ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )
13	12	2ralbidv 2989	. . . 4 |- ( f = S -> ( A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) ) <-> A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )
14	6, 13	anbi12d 747	. . 3 |- ( f = S -> ( ( ( f ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) ) ) <-> ( ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) ) )
15		df-st 29070	. . 3 |- States = { f e. ( ( 0 [,] 1 ) ^m CH ) | ( ( f ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( f ` ( x vH y ) ) = ( ( f ` x ) + ( f ` y ) ) ) ) }
16	14, 15	elrab2 3366	. 2 |- ( S e. States <-> ( S e. ( ( 0 [,] 1 ) ^m CH ) /\ ( ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) ) )
17		3anass 1042	. 2 |- ( ( S : CH --> ( 0 [,] 1 ) /\ ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) <-> ( S : CH --> ( 0 [,] 1 ) /\ ( ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) ) )
18	4, 16, 17	3bitr4i 292	1 |- ( S e. States <-> ( S : CH --> ( 0 [,] 1 ) /\ ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   [,]cicc 12178   ~Hchil 27776   CHcch 27786   _|_cort 27787   vH chj 27790   Statescst 27819

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-sh 28064  df-ch 28078  df-st 29070

This theorem is referenced by:  sticl  29074  sthil  29093  stj  29094  strlem3a  29111