Theorem abvtrivd 16848

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 2434	. 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 9373	. . . . 5 |- 0 e. RR
12		1re 9372	. . . . 5 |- 1 e. RR
13	11, 12	keepel 3845	. . . 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 5855	. 2 |- ( ph -> F : B --> RR )
17	3, 8	rng0cl 16601	. . 3 |- ( R e. Ring -> .0. e. B )
18		iftrue 3785	. . . 4 |- ( x = .0. -> if ( x = .0. , 0 , 1 ) = 0 )
19		c0ex 9367	. . . 4 |- 0 e. _V
20	18, 15, 19	fvmpt 5762	. . 3 |- ( .0. e. B -> ( F ` .0. ) = 0 )
21	10, 17, 20	3syl 20	. 2 |- ( ph -> ( F ` .0. ) = 0 )
22		0lt1 9849	. . 3 |- 0 < 1
23		eqeq1 2439	. . . . . . 7 |- ( x = y -> ( x = .0. <-> y = .0. ) )
24	23	ifbid 3799	. . . . . 6 |- ( x = y -> if ( x = .0. , 0 , 1 ) = if ( y = .0. , 0 , 1 ) )
25		1ex 9368	. . . . . . 7 |- 1 e. _V
26	19, 25	ifex 3846	. . . . . 6 |- if ( y = .0. , 0 , 1 ) e. _V
27	24, 15, 26	fvmpt 5762	. . . . 5 |- ( y e. B -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
28		ifnefalse 3789	. . . . 5 |- ( y =/= .0. -> if ( y = .0. , 0 , 1 ) = 1 )
29	27, 28	sylan9eq 2485	. . . 4 |- ( ( y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
30	29	3adant1 999	. . 3 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> ( F ` y ) = 1 )
31	22, 30	syl5breqr 4316	. 2 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> 0 < ( F ` y ) )
32		1t1e1 10456	. . . 4 |- ( 1 x. 1 ) = 1
33	32	eqcomi 2437	. . 3 |- 1 = ( 1 x. 1 )
34	10	3ad2ant1 1002	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Ring )
35		simp2l 1007	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y e. B )
36		simp3l 1009	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z e. B )
37	3, 6	rngcl 16593	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y .x. z ) e. B )
38	34, 35, 36, 37	syl3anc 1211	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y .x. z ) e. B )
39		eqeq1 2439	. . . . . . 7 |- ( x = ( y .x. z ) -> ( x = .0. <-> ( y .x. z ) = .0. ) )
40	39	ifbid 3799	. . . . . 6 |- ( x = ( y .x. z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
41	19, 25	ifex 3846	. . . . . 6 |- if ( ( y .x. z ) = .0. , 0 , 1 ) e. _V
42	40, 15, 41	fvmpt 5762	. . . . 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 2614	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. ( y .x. z ) = .0. )
46		iffalse 3787	. . . . 5 |- ( -. ( y .x. z ) = .0. -> if ( ( y .x. z ) = .0. , 0 , 1 ) = 1 )
47	45, 46	syl 16	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y .x. z ) = .0. , 0 , 1 ) = 1 )
48	43, 47	eqtrd 2465	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = 1 )
49	35, 27	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
50		simp2r 1008	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y =/= .0. )
51	50	neneqd 2614	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. y = .0. )
52		iffalse 3787	. . . . . 6 |- ( -. y = .0. -> if ( y = .0. , 0 , 1 ) = 1 )
53	51, 52	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( y = .0. , 0 , 1 ) = 1 )
54	49, 53	eqtrd 2465	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = 1 )
55		eqeq1 2439	. . . . . . . 8 |- ( x = z -> ( x = .0. <-> z = .0. ) )
56	55	ifbid 3799	. . . . . . 7 |- ( x = z -> if ( x = .0. , 0 , 1 ) = if ( z = .0. , 0 , 1 ) )
57	19, 25	ifex 3846	. . . . . . 7 |- if ( z = .0. , 0 , 1 ) e. _V
58	56, 15, 57	fvmpt 5762	. . . . . 6 |- ( z e. B -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
59	36, 58	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = if ( z = .0. , 0 , 1 ) )
60		simp3r 1010	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z =/= .0. )
61	60	neneqd 2614	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. z = .0. )
62		iffalse 3787	. . . . . 6 |- ( -. z = .0. -> if ( z = .0. , 0 , 1 ) = 1 )
63	61, 62	syl 16	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( z = .0. , 0 , 1 ) = 1 )
64	59, 63	eqtrd 2465	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = 1 )
65	54, 64	oveq12d 6098	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) x. ( F ` z ) ) = ( 1 x. 1 ) )
66	33, 48, 65	3eqtr4a 2491	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
67		breq1 4283	. . . . . 6 |- ( 0 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 0 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
68		breq1 4283	. . . . . 6 |- ( 1 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 1 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
69		0le2 10399	. . . . . 6 |- 0 <_ 2
70		1le2 10522	. . . . . 6 |- 1 <_ 2
71	67, 68, 69, 70	keephyp 3842	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2
72		df-2 10367	. . . . 5 |- 2 = ( 1 + 1 )
73	71, 72	breqtri 4303	. . . 4 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ ( 1 + 1 )
74	73	a1i 11	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ ( 1 + 1 ) )
75		rnggrp 16585	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
76	10, 75	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
77	76	3ad2ant1 1002	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Grp )
78		eqid 2433	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
79	3, 78	grpcl 15530	. . . . 5 |- ( ( R e. Grp /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
80	77, 35, 36, 79	syl3anc 1211	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( +g ` R ) z ) e. B )
81		eqeq1 2439	. . . . . 6 |- ( x = ( y ( +g ` R ) z ) -> ( x = .0. <-> ( y ( +g ` R ) z ) = .0. ) )
82	81	ifbid 3799	. . . . 5 |- ( x = ( y ( +g ` R ) z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
83	19, 25	ifex 3846	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) e. _V
84	82, 15, 83	fvmpt 5762	. . . 4 |- ( ( y ( +g ` R ) z ) e. B -> ( F ` ( y ( +g ` R ) z ) ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
85	80, 84	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 ) )
86	54, 64	oveq12d 6098	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) + ( F ` z ) ) = ( 1 + 1 ) )
87	74, 85, 86	3brtr4d 4310	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y ( +g ` R ) z ) ) <_ ( ( F ` y ) + ( F ` z ) ) )
88	2, 4, 5, 7, 9, 10, 16, 21, 31, 66, 87	isabvd 16828	1 |- ( ph -> F e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 958   = wceq 1362   e. wcel 1755   =/= wne 2596   ifcif 3779   class class class wbr 4280   e. cmpt 4338   ` cfv 5406  (class class class)co 6080   RRcr 9268   0cc0 9269   1c1 9270   + caddc 9272   x. cmul 9274   < clt 9405   <_ cle 9406   2c2 10358   Basecbs 14156   +g cplusg 14220   .rcmulr 14221   0gc0g 14360   Grpcgrp 15392   Ringcrg 16576  AbsValcabv 16824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-2 10367  df-ico 11293  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-plusg 14233  df-0g 14362  df-mnd 15397  df-grp 15524  df-minusg 15525  df-mgp 16565  df-rng 16579  df-abv 16825

This theorem is referenced by:  abvtriv  16849  abvn0b  17295