Theorem abveq0 18826

Description: The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)

Hypotheses

Ref	Expression
abvf.a	|- A = (AbsVal ` R )
abvf.b	|- B = ( Base ` R )
abveq0.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
abveq0	|- ( ( F e. A /\ X e. B ) -> ( ( F ` X ) = 0 <-> X = .0. ) )

Proof of Theorem abveq0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abvf.a	. . . . . . 7 |- A = (AbsVal ` R )
2	1	abvrcl 18821	. . . . . 6 |- ( F e. A -> R e. Ring )
3		abvf.b	. . . . . . 7 |- B = ( Base ` R )
4		eqid 2622	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
5		eqid 2622	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
6		abveq0.z	. . . . . . 7 |- .0. = ( 0g ` R )
7	1, 3, 4, 5, 6	isabv 18819	. . . . . 6 |- ( R e. Ring -> ( F e. A <-> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) ) )
8	2, 7	syl 17	. . . . 5 |- ( F e. A -> ( F e. A <-> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) ) )
9	8	ibi 256	. . . 4 |- ( F e. A -> ( F : B --> ( 0 [,) +oo ) /\ A. x e. B ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) ) )
10	9	simprd 479	. . 3 |- ( F e. A -> A. x e. B ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) )
11		simpl 473	. . . 4 |- ( ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) -> ( ( F ` x ) = 0 <-> x = .0. ) )
12	11	ralimi 2952	. . 3 |- ( A. x e. B ( ( ( F ` x ) = 0 <-> x = .0. ) /\ A. y e. B ( ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) /\ ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) ) ) -> A. x e. B ( ( F ` x ) = 0 <-> x = .0. ) )
13	10, 12	syl 17	. 2 |- ( F e. A -> A. x e. B ( ( F ` x ) = 0 <-> x = .0. ) )
14		fveq2 6191	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
15	14	eqeq1d 2624	. . . 4 |- ( x = X -> ( ( F ` x ) = 0 <-> ( F ` X ) = 0 ) )
16		eqeq1 2626	. . . 4 |- ( x = X -> ( x = .0. <-> X = .0. ) )
17	15, 16	bibi12d 335	. . 3 |- ( x = X -> ( ( ( F ` x ) = 0 <-> x = .0. ) <-> ( ( F ` X ) = 0 <-> X = .0. ) ) )
18	17	rspccva 3308	. 2 |- ( ( A. x e. B ( ( F ` x ) = 0 <-> x = .0. ) /\ X e. B ) -> ( ( F ` X ) = 0 <-> X = .0. ) )
19	13, 18	sylan 488	1 |- ( ( F e. A /\ X e. B ) -> ( ( F ` X ) = 0 <-> X = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   x. cmul 9941   +oocpnf 10071   <_ cle 10075   [,)cico 12177   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Ringcrg 18547  AbsValcabv 18816

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

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  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-abv 18817

This theorem is referenced by:  abvne0  18827  abv0  18831  abvmet  22380