Theorem abvtrivd 17616

Description: The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
abvtriv.a	|- A = (AbsVal ` R )
abvtriv.b	|- B = ( Base ` R )
abvtriv.z	|- .0. = ( 0g ` R )
abvtriv.f	|- F = ( x e. B |-> if ( x = .0. , 0 , 1 ) )
abvtrivd.1	|- .x. = ( .r ` R )
abvtrivd.2	|- ( ph -> R e. Ring )
abvtrivd.3	|- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) =/= .0. )

Assertion

Ref	Expression
abvtrivd	|- ( ph -> F e. A )

Distinct variable groups:   x, .0.   y, z, F   x, y, z, ph   x, R, y, z   x, .x.   x, B

Allowed substitution hints:   A( x, y, z)   B( y, z)   .x. ( y, z)   F( x)   .0. ( y, z)

Proof of Theorem abvtrivd

Step	Hyp	Ref	Expression
1		abvtriv.a	. . 3 |- A = (AbsVal ` R )
2	1	a1i 11	. 2 |- ( ph -> A = (AbsVal ` R ) )
3		abvtriv.b	. . 3 |- B = ( Base ` R )
4	3	a1i 11	. 2 |- ( ph -> B = ( Base ` R ) )
5		eqidd 2458	. 2 |- ( ph -> ( +g ` R ) = ( +g ` R ) )
6		abvtrivd.1	. . 3 |- .x. = ( .r ` R )
7	6	a1i 11	. 2 |- ( ph -> .x. = ( .r ` R ) )
8		abvtriv.z	. . 3 |- .0. = ( 0g ` R )
9	8	a1i 11	. 2 |- ( ph -> .0. = ( 0g ` R ) )
10		abvtrivd.2	. 2 |- ( ph -> R e. Ring )
11		0re 9613	. . . . 5 |- 0 e. RR
12		1re 9612	. . . . 5 |- 1 e. RR
13	11, 12	keepel 4012	. . . 4 |- if ( x = .0. , 0 , 1 ) e. RR
14	13	a1i 11	. . 3 |- ( ( ph /\ x e. B ) -> if ( x = .0. , 0 , 1 ) e. RR )
15		abvtriv.f	. . 3 |- F = ( x e. B |-> if ( x = .0. , 0 , 1 ) )
16	14, 15	fmptd 6056	. 2 |- ( ph -> F : B --> RR )
17	3, 8	ring0cl 17347	. . 3 |- ( R e. Ring -> .0. e. B )
18		iftrue 3950	. . . 4 |- ( x = .0. -> if ( x = .0. , 0 , 1 ) = 0 )
19		c0ex 9607	. . . 4 |- 0 e. _V
20	18, 15, 19	fvmpt 5956	. . 3 |- ( .0. e. B -> ( F ` .0. ) = 0 )
21	10, 17, 20	3syl 20	. 2 |- ( ph -> ( F ` .0. ) = 0 )
22		0lt1 10096	. . 3 |- 0 < 1
23		eqeq1 2461	. . . . . . 7 |- ( x = y -> ( x = .0. <-> y = .0. ) )
24	23	ifbid 3966	. . . . . 6 |- ( x = y -> if ( x = .0. , 0 , 1 ) = if ( y = .0. , 0 , 1 ) )
25		1ex 9608	. . . . . . 7 |- 1 e. _V
26	19, 25	ifex 4013	. . . . . 6 |- if ( y = .0. , 0 , 1 ) e. _V
27	24, 15, 26	fvmpt 5956	. . . . 5 |- ( y e. B -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
28		ifnefalse 3956	. . . . 5 |- ( y =/= .0. -> if ( y = .0. , 0 , 1 ) = 1 )
29	27, 28	sylan9eq 2518	. . . 4 |- ( ( y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
30	29	3adant1 1014	. . 3 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
31	22, 30	syl5breqr 4492	. 2 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> 0 < ( F ` y ) )
32		1t1e1 10704	. . . 4 |- ( 1 x. 1 ) = 1
33	32	eqcomi 2470	. . 3 |- 1 = ( 1 x. 1 )
34	10	3ad2ant1 1017	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Ring )
35		simp2l 1022	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y e. B )
36		simp3l 1024	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z e. B )
37	3, 6	ringcl 17339	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y .x. z ) e. B )
38	34, 35, 36, 37	syl3anc 1228	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) e. B )
39		eqeq1 2461	. . . . . . 7 |- ( x = ( y .x. z ) -> ( x = .0. <-> ( y .x. z ) = .0. ) )
40	39	ifbid 3966	. . . . . 6 |- ( x = ( y .x. z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
41	19, 25	ifex 4013	. . . . . 6 |- if ( ( y .x. z ) = .0. , 0 , 1 ) e. _V
42	40, 15, 41	fvmpt 5956	. . . . 5 |- ( ( y .x. z ) e. B -> ( F ` ( y .x. z ) ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
43	38, 42	syl 16	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
44		abvtrivd.3	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) =/= .0. )
45	44	neneqd 2659	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. ( y .x. z ) = .0. )
46	45	iffalsed 3955	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y .x. z ) = .0. , 0 , 1 ) = 1 )
47	43, 46	eqtrd 2498	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = 1 )
48	35, 27	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
49		simp2r 1023	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y =/= .0. )
50	49	neneqd 2659	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. y = .0. )
51	50	iffalsed 3955	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( y = .0. , 0 , 1 ) = 1 )
52	48, 51	eqtrd 2498	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = 1 )
53		eqeq1 2461	. . . . . . . 8 |- ( x = z -> ( x = .0. <-> z = .0. ) )
54	53	ifbid 3966	. . . . . . 7 |- ( x = z -> if ( x = .0. , 0 , 1 ) = if ( z = .0. , 0 , 1 ) )
55	19, 25	ifex 4013	. . . . . . 7 |- if ( z = .0. , 0 , 1 ) e. _V
56	54, 15, 55	fvmpt 5956	. . . . . 6 |- ( z e. B -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
57	36, 56	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
58		simp3r 1025	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z =/= .0. )
59	58	neneqd 2659	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. z = .0. )
60	59	iffalsed 3955	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( z = .0. , 0 , 1 ) = 1 )
61	57, 60	eqtrd 2498	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = 1 )
62	52, 61	oveq12d 6314	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) x. ( F ` z ) ) = ( 1 x. 1 ) )
63	33, 47, 62	3eqtr4a 2524	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
64		breq1 4459	. . . . . 6 |- ( 0 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 0 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
65		breq1 4459	. . . . . 6 |- ( 1 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 1 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
66		0le2 10647	. . . . . 6 |- 0 <_ 2
67		1le2 10770	. . . . . 6 |- 1 <_ 2
68	64, 65, 66, 67	keephyp 4009	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2
69		df-2 10615	. . . . 5 |- 2 = ( 1 + 1 )
70	68, 69	breqtri 4479	. . . 4 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ ( 1 + 1 )
71	70	a1i 11	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ ( 1 + 1 ) )
72		ringgrp 17330	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
73	10, 72	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
74	73	3ad2ant1 1017	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Grp )
75		eqid 2457	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
76	3, 75	grpcl 16190	. . . . 5 |- ( ( R e. Grp /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
77	74, 35, 36, 76	syl3anc 1228	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( +g ` R ) z ) e. B )
78		eqeq1 2461	. . . . . 6 |- ( x = ( y ( +g ` R ) z ) -> ( x = .0. <-> ( y ( +g ` R ) z ) = .0. ) )
79	78	ifbid 3966	. . . . 5 |- ( x = ( y ( +g ` R ) z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
80	19, 25	ifex 4013	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) e. _V
81	79, 15, 80	fvmpt 5956	. . . 4 |- ( ( y ( +g ` R ) z ) e. B -> ( F ` ( y ( +g ` R ) z ) ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
82	77, 81	syl 16	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y ( +g ` R ) z ) ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
83	52, 61	oveq12d 6314	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) + ( F ` z ) ) = ( 1 + 1 ) )
84	71, 82, 83	3brtr4d 4486	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y ( +g ` R ) z ) ) <_ ( ( F ` y ) + ( F ` z ) ) )
85	2, 4, 5, 7, 9, 10, 16, 21, 31, 63, 84	isabvd 17596	1 |- ( ph -> F e. A )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   ifcif 3944   class class class wbr 4456   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   < clt 9645   <_ cle 9646   2c2 10606   Basecbs 14644   +g cplusg 14712   .rcmulr 14713   0gc0g 14857   Grpcgrp 16180   Ringcrg 17325  AbsValcabv 17592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ico 11560  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-mgp 17269  df-ring 17327  df-abv 17593

This theorem is referenced by:  abvtriv  17617  abvn0b  18078