Theorem abvtrivd 17360

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 2468	. 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 9608	. . . . 5 |- 0 e. RR
12		1re 9607	. . . . 5 |- 1 e. RR
13	11, 12	keepel 4013	. . . 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 17092	. . 3 |- ( R e. Ring -> .0. e. B )
18		iftrue 3951	. . . 4 |- ( x = .0. -> if ( x = .0. , 0 , 1 ) = 0 )
19		c0ex 9602	. . . 4 |- 0 e. _V
20	18, 15, 19	fvmpt 5957	. . 3 |- ( .0. e. B -> ( F ` .0. ) = 0 )
21	10, 17, 20	3syl 20	. 2 |- ( ph -> ( F ` .0. ) = 0 )
22		0lt1 10087	. . 3 |- 0 < 1
23		eqeq1 2471	. . . . . . 7 |- ( x = y -> ( x = .0. <-> y = .0. ) )
24	23	ifbid 3967	. . . . . 6 |- ( x = y -> if ( x = .0. , 0 , 1 ) = if ( y = .0. , 0 , 1 ) )
25		1ex 9603	. . . . . . 7 |- 1 e. _V
26	19, 25	ifex 4014	. . . . . 6 |- if ( y = .0. , 0 , 1 ) e. _V
27	24, 15, 26	fvmpt 5957	. . . . 5 |- ( y e. B -> ( F ` y ) = if ( y = .0. , 0 , 1 ) )
28		ifnefalse 3957	. . . . 5 |- ( y =/= .0. -> if ( y = .0. , 0 , 1 ) = 1 )
29	27, 28	sylan9eq 2528	. . . 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 4489	. 2 |- ( ( ph /\ y e. B /\ y =/= .0. ) -> 0 < ( F ` y ) )
32		1t1e1 10695	. . . 4 |- ( 1 x. 1 ) = 1
33	32	eqcomi 2480	. . 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 17084	. . . . . 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 2471	. . . . . . 7 |- ( x = ( y .x. z ) -> ( x = .0. <-> ( y .x. z ) = .0. ) )
40	39	ifbid 3967	. . . . . 6 |- ( x = ( y .x. z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y .x. z ) = .0. , 0 , 1 ) )
41	19, 25	ifex 4014	. . . . . 6 |- if ( ( y .x. z ) = .0. , 0 , 1 ) e. _V
42	40, 15, 41	fvmpt 5957	. . . . 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 2669	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. ( y .x. z ) = .0. )
46		iffalse 3954	. . . . 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 2508	. . 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 1023	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> y =/= .0. )
51	50	neneqd 2669	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. y = .0. )
52		iffalse 3954	. . . . . 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 2508	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` y ) = 1 )
55		eqeq1 2471	. . . . . . . 8 |- ( x = z -> ( x = .0. <-> z = .0. ) )
56	55	ifbid 3967	. . . . . . 7 |- ( x = z -> if ( x = .0. , 0 , 1 ) = if ( z = .0. , 0 , 1 ) )
57	19, 25	ifex 4014	. . . . . . 7 |- if ( z = .0. , 0 , 1 ) e. _V
58	56, 15, 57	fvmpt 5957	. . . . . 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 1025	. . . . . . 7 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> z =/= .0. )
61	60	neneqd 2669	. . . . . 6 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> -. z = .0. )
62		iffalse 3954	. . . . . 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 2508	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` z ) = 1 )
65	54, 64	oveq12d 6313	. . 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 2534	. 2 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( F ` ( y .x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
67		breq1 4456	. . . . . 6 |- ( 0 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 0 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
68		breq1 4456	. . . . . 6 |- ( 1 = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) -> ( 1 <_ 2 <-> if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2 ) )
69		0le2 10638	. . . . . 6 |- 0 <_ 2
70		1le2 10761	. . . . . 6 |- 1 <_ 2
71	67, 68, 69, 70	keephyp 4010	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) <_ 2
72		df-2 10606	. . . . 5 |- 2 = ( 1 + 1 )
73	71, 72	breqtri 4476	. . . 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		ringgrp 17075	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
76	10, 75	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
77	76	3ad2ant1 1017	. . . . 5 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> R e. Grp )
78		eqid 2467	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
79	3, 78	grpcl 15935	. . . . 5 |- ( ( R e. Grp /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
80	77, 35, 36, 79	syl3anc 1228	. . . 4 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( y ( +g ` R ) z ) e. B )
81		eqeq1 2471	. . . . . 6 |- ( x = ( y ( +g ` R ) z ) -> ( x = .0. <-> ( y ( +g ` R ) z ) = .0. ) )
82	81	ifbid 3967	. . . . 5 |- ( x = ( y ( +g ` R ) z ) -> if ( x = .0. , 0 , 1 ) = if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) )
83	19, 25	ifex 4014	. . . . 5 |- if ( ( y ( +g ` R ) z ) = .0. , 0 , 1 ) e. _V
84	82, 15, 83	fvmpt 5957	. . . 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 6313	. . 3 |- ( ( ph /\ ( y e. B /\ y =/= .0. ) /\ ( z e. B /\ z =/= .0. ) ) -> ( ( F ` y ) + ( F ` z ) ) = ( 1 + 1 ) )
87	74, 85, 86	3brtr4d 4483	. 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 17340	1 |- ( ph -> F e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   ifcif 3945   class class class wbr 4453   |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505   + caddc 9507   x. cmul 9509   < clt 9640   <_ cle 9641   2c2 10597   Basecbs 14507   +g cplusg 14572   .rcmulr 14573   0gc0g 14712   Grpcgrp 15925   Ringcrg 17070  AbsValcabv 17336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ico 11547  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-mgp 17014  df-ring 17072  df-abv 17337

This theorem is referenced by:  abvtriv  17361  abvn0b  17821