Theorem abvtrivd 18068

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 2452	. 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 9643	. . . . 5 |- 0 e. RR
12		1re 9642	. . . . 5 |- 1 e. RR
13	11, 12	keepel 3948	. . . 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 6046	. 2 |- ( ph -> F : B --> RR )
17	3, 8	ring0cl 17802	. . 3 |- ( R e. Ring -> .0. e. B )
18		iftrue 3887	. . . 4 |- ( x = .0. -> if ( x = .0. , 0 , 1 ) = 0 )
19		c0ex 9637	. . . 4 |- 0 e. _V
20	18, 15, 19	fvmpt 5948	. . 3 |- ( .0. e. B -> ( F ` .0. ) = 0 )
21	10, 17, 20	3syl 18	. 2 |- ( ph -> ( F ` .0. ) = 0 )
22		0lt1 10136	. . 3 |- 0 < 1
23		eqeq1 2455	. . . . . . 7 |- ( x = y -> ( x = .0. <-> y = .0. ) )
24	23	ifbid 3903	. . . . . 6 |- ( x = y -> if ( x = .0. , 0 , 1 ) = if ( y = .0. , 0 , 1 ) )
25		1ex 9638	. . . . . . 7 |- 1 e. _V
26	19, 25	ifex 3949	. . . . . 6 |- if ( y = .0. , 0 , 1 ) e. _V
27	24, 15, 26	fvmpt 5948	. . . . 5 |- ( y e. B -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
28		ifnefalse 3893	. . . . 5 |- ( y =/= .0. -> if ( y = .0. , 0 , 1 ) = 1 )
29	27, 28	sylan9eq 2505	. . . 4 |- ( ( y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
30	29	3adant1 1026	. . 3 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
31	22, 30	syl5breqr 4439	. 2 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> 0 < ( F ` y ) )
32		1t1e1 10757	. . . 4 |- ( 1 x. 1 ) = 1
33	32	eqcomi 2460	. . 3 |- 1 = ( 1 x. 1 )
34	10	3ad2ant1 1029	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Ring )
35		simp2l 1034	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y e. B )
36		simp3l 1036	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z e. B )
37	3, 6	ringcl 17794	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y .x. z ) e. B )
38	34, 35, 36, 37	syl3anc 1268	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) e. B )
39		eqeq1 2455	. . . . . . 7 |- ( x = ( y .x. z ) -> ( x = .0. <-> ( y .x. z ) = .0. ) )
40	39	ifbid 3903	. . . . . 6 |- ( x = ( y .x. z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
41	19, 25	ifex 3949	. . . . . 6 |- if ( ( y .x. z ) = .0. , 0 , 1 ) e. _V
42	40, 15, 41	fvmpt 5948	. . . . 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 2629	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. ( y .x. z ) = .0. )
46	45	iffalsed 3892	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y .x. z ) = .0. , 0 , 1 ) = 1 )
47	43, 46	eqtrd 2485	. . 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 1035	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y =/= .0. )
50	49	neneqd 2629	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. y = .0. )
51	50	iffalsed 3892	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( y = .0. , 0 , 1 ) = 1 )
52	48, 51	eqtrd 2485	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = 1 )
53		eqeq1 2455	. . . . . . . 8 |- ( x = z -> ( x = .0. <-> z = .0. ) )
54	53	ifbid 3903	. . . . . . 7 |- ( x = z -> if ( x = .0. , 0 , 1 ) = if ( z = .0. , 0 , 1 ) )
55	19, 25	ifex 3949	. . . . . . 7 |- if ( z = .0. , 0 , 1 ) e. _V
56	54, 15, 55	fvmpt 5948	. . . . . 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 1037	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z =/= .0. )
59	58	neneqd 2629	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. z = .0. )
60	59	iffalsed 3892	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( z = .0. , 0 , 1 ) = 1 )
61	57, 60	eqtrd 2485	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = 1 )
62	52, 61	oveq12d 6308	. . 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 2511	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
64		breq1 4405	. . . . . 6 |- ( 0 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 0 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
65		breq1 4405	. . . . . 6 |- ( 1 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 1 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
66		0le2 10700	. . . . . 6 |- 0 <_ 2
67		1le2 10823	. . . . . 6 |- 1 <_ 2
68	64, 65, 66, 67	keephyp 3945	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2
69		df-2 10668	. . . . 5 |- 2 = ( 1 + 1 )
70	68, 69	breqtri 4426	. . . 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 17785	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
73	10, 72	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
74	73	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Grp )
75		eqid 2451	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
76	3, 75	grpcl 16679	. . . . 5 |- ( ( R e. Grp /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
77	74, 35, 36, 76	syl3anc 1268	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( +g ` R ) z ) e. B )
78		eqeq1 2455	. . . . . 6 |- ( x = ( y ( +g ` R ) z ) -> ( x = .0. <-> ( y ( +g ` R ) z ) = .0. ) )
79	78	ifbid 3903	. . . . 5 |- ( x = ( y ( +g ` R ) z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
80	19, 25	ifex 3949	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) e. _V
81	79, 15, 80	fvmpt 5948	. . . 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 6308	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) + ( F ` z ) ) = ( 1 + 1 ) )
84	71, 82, 83	3brtr4d 4433	. 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 18048	1 |- ( ph -> F e. A )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   ifcif 3881   class class class wbr 4402   |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   < clt 9675   <_ cle 9676   2c2 10659   Basecbs 15121   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   Ringcrg 17780  AbsValcabv 18044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ico 11641  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ring 17782  df-abv 18045

This theorem is referenced by:  abvtriv  18069  abvn0b  18526