MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abvtrivd Structured version   Unicode version

Theorem abvtrivd 15920
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
StepHypRef Expression
1 abvtriv.a . . 3  |-  A  =  (AbsVal `  R )
21a1i 11 . 2  |-  ( ph  ->  A  =  (AbsVal `  R ) )
3 abvtriv.b . . 3  |-  B  =  ( Base `  R
)
43a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  R ) )
5 eqidd 2436 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
6 abvtrivd.1 . . 3  |-  .x.  =  ( .r `  R )
76a1i 11 . 2  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 abvtriv.z . . 3  |-  .0.  =  ( 0g `  R )
98a1i 11 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
10 abvtrivd.2 . 2  |-  ( ph  ->  R  e.  Ring )
11 0re 9083 . . . . 5  |-  0  e.  RR
12 1re 9082 . . . . 5  |-  1  e.  RR
1311, 12keepel 3788 . . . 4  |-  if ( x  =  .0.  , 
0 ,  1 )  e.  RR
1413a1i 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 ) )
1614, 15fmptd 5885 . 2  |-  ( ph  ->  F : B --> RR )
173, 8rng0cl 15677 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  B )
18 iftrue 3737 . . . 4  |-  ( x  =  .0.  ->  if ( x  =  .0.  ,  0 ,  1 )  =  0 )
19 c0ex 9077 . . . 4  |-  0  e.  _V
2018, 15, 19fvmpt 5798 . . 3  |-  (  .0. 
e.  B  ->  ( F `  .0.  )  =  0 )
2110, 17, 203syl 19 . 2  |-  ( ph  ->  ( F `  .0.  )  =  0 )
22 0lt1 9542 . . 3  |-  0  <  1
23 eqeq1 2441 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  .0.  <->  y  =  .0.  ) )
2423ifbid 3749 . . . . . 6  |-  ( x  =  y  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( y  =  .0.  ,  0 ,  1 ) )
25 1ex 9078 . . . . . . 7  |-  1  e.  _V
2619, 25ifex 3789 . . . . . 6  |-  if ( y  =  .0.  , 
0 ,  1 )  e.  _V
2724, 15, 26fvmpt 5798 . . . . 5  |-  ( y  e.  B  ->  ( F `  y )  =  if ( y  =  .0.  ,  0 ,  1 ) )
28 ifnefalse 3739 . . . . 5  |-  ( y  =/=  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
2927, 28sylan9eq 2487 . . . 4  |-  ( ( y  e.  B  /\  y  =/=  .0.  )  -> 
( F `  y
)  =  1 )
30293adant1 975 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  ( F `  y )  =  1 )
3122, 30syl5breqr 4240 . 2  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  0  <  ( F `  y )
)
32 1t1e1 10118 . . . 4  |-  ( 1  x.  1 )  =  1
3332eqcomi 2439 . . 3  |-  1  =  ( 1  x.  1 )
34103ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Ring )
35 simp2l 983 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  e.  B
)
36 simp3l 985 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  e.  B
)
373, 6rngcl 15669 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y  .x.  z )  e.  B )
3834, 35, 36, 37syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y  .x.  z )  e.  B
)
39 eqeq1 2441 . . . . . . 7  |-  ( x  =  ( y  .x.  z )  ->  (
x  =  .0.  <->  ( y  .x.  z )  =  .0.  ) )
4039ifbid 3749 . . . . . 6  |-  ( x  =  ( y  .x.  z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 ) )
4119, 25ifex 3789 . . . . . 6  |-  if ( ( y  .x.  z
)  =  .0.  , 
0 ,  1 )  e.  _V
4240, 15, 41fvmpt 5798 . . . . 5  |-  ( ( y  .x.  z )  e.  B  ->  ( F `  ( y  .x.  z ) )  =  if ( ( y 
.x.  z )  =  .0.  ,  0 ,  1 ) )
4338, 42syl 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.  )
4544neneqd 2614 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  ( y  .x.  z )  =  .0.  )
46 iffalse 3738 . . . . 5  |-  ( -.  ( y  .x.  z
)  =  .0.  ->  if ( ( y  .x.  z )  =  .0. 
,  0 ,  1 )  =  1 )
4745, 46syl 16 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 )  =  1 )
4843, 47eqtrd 2467 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  1 )
4935, 27syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  if ( y  =  .0. 
,  0 ,  1 ) )
50 simp2r 984 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  =/=  .0.  )
5150neneqd 2614 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  y  =  .0.  )
52 iffalse 3738 . . . . . 6  |-  ( -.  y  =  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
5351, 52syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( y  =  .0.  ,  0 ,  1 )  =  1 )
5449, 53eqtrd 2467 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  1 )
55 eqeq1 2441 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  .0.  <->  z  =  .0.  ) )
5655ifbid 3749 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5719, 25ifex 3789 . . . . . . 7  |-  if ( z  =  .0.  , 
0 ,  1 )  e.  _V
5856, 15, 57fvmpt 5798 . . . . . 6  |-  ( z  e.  B  ->  ( F `  z )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5936, 58syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  if ( z  =  .0. 
,  0 ,  1 ) )
60 simp3r 986 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  =/=  .0.  )
6160neneqd 2614 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  z  =  .0.  )
62 iffalse 3738 . . . . . 6  |-  ( -.  z  =  .0.  ->  if ( z  =  .0. 
,  0 ,  1 )  =  1 )
6361, 62syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( z  =  .0.  ,  0 ,  1 )  =  1 )
6459, 63eqtrd 2467 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  1 )
6554, 64oveq12d 6091 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( 1  x.  1 ) )
6633, 48, 653eqtr4a 2493 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
67 breq1 4207 . . . . . 6  |-  ( 0  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
0  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
68 breq1 4207 . . . . . 6  |-  ( 1  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
1  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
69 2re 10061 . . . . . . 7  |-  2  e.  RR
70 2pos 10074 . . . . . . 7  |-  0  <  2
7111, 69, 70ltleii 9188 . . . . . 6  |-  0  <_  2
72 1lt2 10134 . . . . . . 7  |-  1  <  2
7312, 69, 72ltleii 9188 . . . . . 6  |-  1  <_  2
7467, 68, 71, 73keephyp 3785 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2
75 df-2 10050 . . . . 5  |-  2  =  ( 1  +  1 )
7674, 75breqtri 4227 . . . 4  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
( 1  +  1 )
7776a1i 11 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  <_  (
1  +  1 ) )
78 rnggrp 15661 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
7910, 78syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
80793ad2ant1 978 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Grp )
81 eqid 2435 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
823, 81grpcl 14810 . . . . 5  |-  ( ( R  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  R ) z )  e.  B )
8380, 35, 36, 82syl3anc 1184 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( +g  `  R ) z )  e.  B
)
84 eqeq1 2441 . . . . . 6  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  =  .0.  <->  ( y
( +g  `  R ) z )  =  .0.  ) )
8584ifbid 3749 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 ) )
8619, 25ifex 3789 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  e. 
_V
8785, 15, 86fvmpt 5798 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( F `  ( y ( +g  `  R ) z ) )  =  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 ) )
8883, 87syl 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 ) )
8954, 64oveq12d 6091 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  +  ( F `  z
) )  =  ( 1  +  1 ) )
9077, 88, 893brtr4d 4234 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y ( +g  `  R ) z ) )  <_  ( ( F `  y )  +  ( F `  z ) ) )
912, 4, 5, 7, 9, 10, 16, 21, 31, 66, 90isabvd 15900 1  |-  ( ph  ->  F  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113   2c2 10041   Basecbs 13461   +g cplusg 13521   .rcmulr 13522   0gc0g 13715   Grpcgrp 14677   Ringcrg 15652  AbsValcabv 15896
This theorem is referenced by:  abvtriv  15921  abvn0b  16354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ico 10914  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mgp 15641  df-rng 15655  df-abv 15897
  Copyright terms: Public domain W3C validator