Theorem signspval 30629

Description: The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)

Hypothesis

Ref	Expression
signsw.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )

Assertion

Ref	Expression
signspval	|- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )

Distinct variable groups:   a, b, X   Y, a, b

Allowed substitution hints:   .+^ ( a, b)

Proof of Theorem signspval

Step	Hyp	Ref	Expression
1		ifcl 4130	. 2 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> if ( Y = 0 , X , Y ) e. { -u 1 , 0 , 1 } )
2		ifeq1 4090	. . 3 |- ( a = X -> if ( b = 0 , a , b ) = if ( b = 0 , X , b ) )
3		eqeq1 2626	. . . 4 |- ( b = Y -> ( b = 0 <-> Y = 0 ) )
4		id 22	. . . 4 |- ( b = Y -> b = Y )
5	3, 4	ifbieq2d 4111	. . 3 |- ( b = Y -> if ( b = 0 , X , b ) = if ( Y = 0 , X , Y ) )
6		signsw.p	. . 3 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
7	2, 5, 6	ovmpt2g 6795	. 2 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } /\ if ( Y = 0 , X , Y ) e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
8	1, 7	mpd3an3 1425	1 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   {ctp 4181  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   -ucneg 10267

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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  signsw0glem  30630  signswmnd  30634  signswrid  30635  signswlid  30636  signswn0  30637  signswch  30638