Theorem signswn0 30637

Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)

Hypotheses

Ref	Expression
signsw.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsw.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }

Assertion

Ref	Expression
signswn0	|- ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) -> ( X .+^ Y ) =/= 0 )

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

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

Proof of Theorem signswn0

Step	Hyp	Ref	Expression
1		signsw.p	. . . 4 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2	1	signspval 30629	. . 3 |- ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
3	2	adantr 481	. 2 |- ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) -> ( X .+^ Y ) = if ( Y = 0 , X , Y ) )
4		neeq1 2856	. . 3 |- ( X = if ( Y = 0 , X , Y ) -> ( X =/= 0 <-> if ( Y = 0 , X , Y ) =/= 0 ) )
5		neeq1 2856	. . 3 |- ( Y = if ( Y = 0 , X , Y ) -> ( Y =/= 0 <-> if ( Y = 0 , X , Y ) =/= 0 ) )
6		simplr 792	. . 3 |- ( ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) /\ Y = 0 ) -> X =/= 0 )
7		simpr 477	. . . 4 |- ( ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) /\ -. Y = 0 ) -> -. Y = 0 )
8	7	neqned 2801	. . 3 |- ( ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) /\ -. Y = 0 ) -> Y =/= 0 )
9	4, 5, 6, 8	ifbothda 4123	. 2 |- ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) -> if ( Y = 0 , X , Y ) =/= 0 )
10	3, 9	eqnetrd 2861	1 |- ( ( ( X e. { -u 1 , 0 , 1 } /\ Y e. { -u 1 , 0 , 1 } ) /\ X =/= 0 ) -> ( X .+^ Y ) =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   -ucneg 10267   ndxcnx 15854   Basecbs 15857   +g cplusg 15941

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-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:  signstfvneq0  30649