Theorem signsw0glem 30630

Description: Neutral element property of .+^. (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
signsw0glem	|- A. u e. { -u 1 , 0 , 1 } ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u )

Distinct variable group:   a, b, u

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

Proof of Theorem signsw0glem

Step	Hyp	Ref	Expression
1		c0ex 10034	. . . . . 6 |- 0 e. _V
2	1	tpid2 4304	. . . . 5 |- 0 e. { -u 1 , 0 , 1 }
3		signsw.p	. . . . . 6 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
4	3	signspval 30629	. . . . 5 |- ( ( 0 e. { -u 1 , 0 , 1 } /\ u e. { -u 1 , 0 , 1 } ) -> ( 0 .+^ u ) = if ( u = 0 , 0 , u ) )
5	2, 4	mpan 706	. . . 4 |- ( u e. { -u 1 , 0 , 1 } -> ( 0 .+^ u ) = if ( u = 0 , 0 , u ) )
6		iftrue 4092	. . . . . 6 |- ( u = 0 -> if ( u = 0 , 0 , u ) = 0 )
7		id 22	. . . . . 6 |- ( u = 0 -> u = 0 )
8	6, 7	eqtr4d 2659	. . . . 5 |- ( u = 0 -> if ( u = 0 , 0 , u ) = u )
9		iffalse 4095	. . . . 5 |- ( -. u = 0 -> if ( u = 0 , 0 , u ) = u )
10	8, 9	pm2.61i 176	. . . 4 |- if ( u = 0 , 0 , u ) = u
11	5, 10	syl6eq 2672	. . 3 |- ( u e. { -u 1 , 0 , 1 } -> ( 0 .+^ u ) = u )
12	3	signspval 30629	. . . . 5 |- ( ( u e. { -u 1 , 0 , 1 } /\ 0 e. { -u 1 , 0 , 1 } ) -> ( u .+^ 0 ) = if ( 0 = 0 , u , 0 ) )
13	2, 12	mpan2 707	. . . 4 |- ( u e. { -u 1 , 0 , 1 } -> ( u .+^ 0 ) = if ( 0 = 0 , u , 0 ) )
14		eqid 2622	. . . . 5 |- 0 = 0
15	14	iftruei 4093	. . . 4 |- if ( 0 = 0 , u , 0 ) = u
16	13, 15	syl6eq 2672	. . 3 |- ( u e. { -u 1 , 0 , 1 } -> ( u .+^ 0 ) = u )
17	11, 16	jca 554	. 2 |- ( u e. { -u 1 , 0 , 1 } -> ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u ) )
18	17	rgen 2922	1 |- A. u e. { -u 1 , 0 , 1 } ( ( 0 .+^ u ) = u /\ ( u .+^ 0 ) = u )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   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  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-tp 4182  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:  signsw0g  30633  signswmnd  30634