Theorem abvtrivd 18840

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 2623	. 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 10040	. . . . 5 |- 0 e. RR
12		1re 10039	. . . . 5 |- 1 e. RR
13	11, 12	keepel 4155	. . . 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 6385	. 2 |- ( ph -> F : B --> RR )
17	3, 8	ring0cl 18569	. . 3 |- ( R e. Ring -> .0. e. B )
18		iftrue 4092	. . . 4 |- ( x = .0. -> if ( x = .0. , 0 , 1 ) = 0 )
19		c0ex 10034	. . . 4 |- 0 e. _V
20	18, 15, 19	fvmpt 6282	. . 3 |- ( .0. e. B -> ( F ` .0. ) = 0 )
21	10, 17, 20	3syl 18	. 2 |- ( ph -> ( F ` .0. ) = 0 )
22		0lt1 10550	. . 3 |- 0 < 1
23		eqeq1 2626	. . . . . . 7 |- ( x = y -> ( x = .0. <-> y = .0. ) )
24	23	ifbid 4108	. . . . . 6 |- ( x = y -> if ( x = .0. , 0 , 1 ) = if ( y = .0. , 0 , 1 ) )
25		1ex 10035	. . . . . . 7 |- 1 e. _V
26	19, 25	ifex 4156	. . . . . 6 |- if ( y = .0. , 0 , 1 ) e. _V
27	24, 15, 26	fvmpt 6282	. . . . 5 |- ( y e. B -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
28		ifnefalse 4098	. . . . 5 |- ( y =/= .0. -> if ( y = .0. , 0 , 1 ) = 1 )
29	27, 28	sylan9eq 2676	. . . 4 |- ( ( y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
30	29	3adant1 1079	. . 3 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
31	22, 30	syl5breqr 4691	. 2 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> 0 < ( F ` y ) )
32		1t1e1 11175	. . . 4 |- ( 1 x. 1 ) = 1
33	32	eqcomi 2631	. . 3 |- 1 = ( 1 x. 1 )
34	10	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Ring )
35		simp2l 1087	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y e. B )
36		simp3l 1089	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z e. B )
37	3, 6	ringcl 18561	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y .x. z ) e. B )
38	34, 35, 36, 37	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) e. B )
39		eqeq1 2626	. . . . . . 7 |- ( x = ( y .x. z ) -> ( x = .0. <-> ( y .x. z ) = .0. ) )
40	39	ifbid 4108	. . . . . 6 |- ( x = ( y .x. z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
41	19, 25	ifex 4156	. . . . . 6 |- if ( ( y .x. z ) = .0. , 0 , 1 ) e. _V
42	40, 15, 41	fvmpt 6282	. . . . 5 |- ( ( y .x. z ) e. B -> ( F ` ( y .x. z ) ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
43	38, 42	syl 17	. . . 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 2799	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. ( y .x. z ) = .0. )
46	45	iffalsed 4097	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y .x. z ) = .0. , 0 , 1 ) = 1 )
47	43, 46	eqtrd 2656	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = 1 )
48	35, 27	syl 17	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
49		simp2r 1088	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y =/= .0. )
50	49	neneqd 2799	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. y = .0. )
51	50	iffalsed 4097	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( y = .0. , 0 , 1 ) = 1 )
52	48, 51	eqtrd 2656	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = 1 )
53		eqeq1 2626	. . . . . . . 8 |- ( x = z -> ( x = .0. <-> z = .0. ) )
54	53	ifbid 4108	. . . . . . 7 |- ( x = z -> if ( x = .0. , 0 , 1 ) = if ( z = .0. , 0 , 1 ) )
55	19, 25	ifex 4156	. . . . . . 7 |- if ( z = .0. , 0 , 1 ) e. _V
56	54, 15, 55	fvmpt 6282	. . . . . 6 |- ( z e. B -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
57	36, 56	syl 17	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
58		simp3r 1090	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z =/= .0. )
59	58	neneqd 2799	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. z = .0. )
60	59	iffalsed 4097	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( z = .0. , 0 , 1 ) = 1 )
61	57, 60	eqtrd 2656	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = 1 )
62	52, 61	oveq12d 6668	. . 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 2682	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
64		breq1 4656	. . . . . 6 |- ( 0 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 0 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
65		breq1 4656	. . . . . 6 |- ( 1 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 1 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
66		0le2 11111	. . . . . 6 |- 0 <_ 2
67		1le2 11241	. . . . . 6 |- 1 <_ 2
68	64, 65, 66, 67	keephyp 4152	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2
69		df-2 11079	. . . . 5 |- 2 = ( 1 + 1 )
70	68, 69	breqtri 4678	. . . 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 18552	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
73	10, 72	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
74	73	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Grp )
75		eqid 2622	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
76	3, 75	grpcl 17430	. . . . 5 |- ( ( R e. Grp /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
77	74, 35, 36, 76	syl3anc 1326	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( +g ` R ) z ) e. B )
78		eqeq1 2626	. . . . . 6 |- ( x = ( y ( +g ` R ) z ) -> ( x = .0. <-> ( y ( +g ` R ) z ) = .0. ) )
79	78	ifbid 4108	. . . . 5 |- ( x = ( y ( +g ` R ) z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
80	19, 25	ifex 4156	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) e. _V
81	79, 15, 80	fvmpt 6282	. . . 4 |- ( ( y ( +g ` R ) z ) e. B -> ( F ` ( y ( +g ` R ) z ) ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
82	77, 81	syl 17	. . 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 6668	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) + ( F ` z ) ) = ( 1 + 1 ) )
84	71, 82, 83	3brtr4d 4685	. 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 18820	1 |- ( ph -> F e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   2c2 11070   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ico 12181  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ring 18549  df-abv 18817

This theorem is referenced by:  abvtriv  18841  abvn0b  19302