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Theorem abvtrivd 15621
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 10 . 2  |-  ( ph  ->  A  =  (AbsVal `  R ) )
3 abvtriv.b . . 3  |-  B  =  ( Base `  R
)
43a1i 10 . 2  |-  ( ph  ->  B  =  ( Base `  R ) )
5 eqidd 2297 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
6 abvtrivd.1 . . 3  |-  .x.  =  ( .r `  R )
76a1i 10 . 2  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 abvtriv.z . . 3  |-  .0.  =  ( 0g `  R )
98a1i 10 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
10 abvtrivd.2 . 2  |-  ( ph  ->  R  e.  Ring )
11 0re 8854 . . . . 5  |-  0  e.  RR
12 1re 8853 . . . . 5  |-  1  e.  RR
1311, 12keepel 3635 . . . 4  |-  if ( x  =  .0.  , 
0 ,  1 )  e.  RR
1413a1i 10 . . 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 5700 . 2  |-  ( ph  ->  F : B --> RR )
173, 8rng0cl 15378 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  B )
18 iftrue 3584 . . . 4  |-  ( x  =  .0.  ->  if ( x  =  .0.  ,  0 ,  1 )  =  0 )
19 c0ex 8848 . . . 4  |-  0  e.  _V
2018, 15, 19fvmpt 5618 . . 3  |-  (  .0. 
e.  B  ->  ( F `  .0.  )  =  0 )
2110, 17, 203syl 18 . 2  |-  ( ph  ->  ( F `  .0.  )  =  0 )
22 0lt1 9312 . . 3  |-  0  <  1
23 eqeq1 2302 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  .0.  <->  y  =  .0.  ) )
2423ifbid 3596 . . . . . 6  |-  ( x  =  y  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( y  =  .0.  ,  0 ,  1 ) )
25 1ex 8849 . . . . . . 7  |-  1  e.  _V
2619, 25ifex 3636 . . . . . 6  |-  if ( y  =  .0.  , 
0 ,  1 )  e.  _V
2724, 15, 26fvmpt 5618 . . . . 5  |-  ( y  e.  B  ->  ( F `  y )  =  if ( y  =  .0.  ,  0 ,  1 ) )
28 ifnefalse 3586 . . . . 5  |-  ( y  =/=  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
2927, 28sylan9eq 2348 . . . 4  |-  ( ( y  e.  B  /\  y  =/=  .0.  )  -> 
( F `  y
)  =  1 )
30293adant1 973 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  ( F `  y )  =  1 )
3122, 30syl5breqr 4075 . 2  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  0  <  ( F `  y )
)
32 1t1e1 9886 . . . 4  |-  ( 1  x.  1 )  =  1
3332eqcomi 2300 . . 3  |-  1  =  ( 1  x.  1 )
34103ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Ring )
35 simp2l 981 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  e.  B
)
36 simp3l 983 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  e.  B
)
373, 6rngcl 15370 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y  .x.  z )  e.  B )
3834, 35, 36, 37syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y  .x.  z )  e.  B
)
39 eqeq1 2302 . . . . . . 7  |-  ( x  =  ( y  .x.  z )  ->  (
x  =  .0.  <->  ( y  .x.  z )  =  .0.  ) )
4039ifbid 3596 . . . . . 6  |-  ( x  =  ( y  .x.  z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 ) )
4119, 25ifex 3636 . . . . . 6  |-  if ( ( y  .x.  z
)  =  .0.  , 
0 ,  1 )  e.  _V
4240, 15, 41fvmpt 5618 . . . . 5  |-  ( ( y  .x.  z )  e.  B  ->  ( F `  ( y  .x.  z ) )  =  if ( ( y 
.x.  z )  =  .0.  ,  0 ,  1 ) )
4338, 42syl 15 . . . 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 2475 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  ( y  .x.  z )  =  .0.  )
46 iffalse 3585 . . . . 5  |-  ( -.  ( y  .x.  z
)  =  .0.  ->  if ( ( y  .x.  z )  =  .0. 
,  0 ,  1 )  =  1 )
4745, 46syl 15 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 )  =  1 )
4843, 47eqtrd 2328 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  1 )
4935, 27syl 15 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  if ( y  =  .0. 
,  0 ,  1 ) )
50 simp2r 982 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  =/=  .0.  )
5150neneqd 2475 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  y  =  .0.  )
52 iffalse 3585 . . . . . 6  |-  ( -.  y  =  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
5351, 52syl 15 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( y  =  .0.  ,  0 ,  1 )  =  1 )
5449, 53eqtrd 2328 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  1 )
55 eqeq1 2302 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  .0.  <->  z  =  .0.  ) )
5655ifbid 3596 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5719, 25ifex 3636 . . . . . . 7  |-  if ( z  =  .0.  , 
0 ,  1 )  e.  _V
5856, 15, 57fvmpt 5618 . . . . . 6  |-  ( z  e.  B  ->  ( F `  z )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5936, 58syl 15 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  if ( z  =  .0. 
,  0 ,  1 ) )
60 simp3r 984 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  =/=  .0.  )
6160neneqd 2475 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  z  =  .0.  )
62 iffalse 3585 . . . . . 6  |-  ( -.  z  =  .0.  ->  if ( z  =  .0. 
,  0 ,  1 )  =  1 )
6361, 62syl 15 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( z  =  .0.  ,  0 ,  1 )  =  1 )
6459, 63eqtrd 2328 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  1 )
6554, 64oveq12d 5892 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( 1  x.  1 ) )
6633, 48, 653eqtr4a 2354 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
67 breq1 4042 . . . . . 6  |-  ( 0  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
0  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
68 breq1 4042 . . . . . 6  |-  ( 1  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
1  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
69 2re 9831 . . . . . . 7  |-  2  e.  RR
70 2pos 9844 . . . . . . 7  |-  0  <  2
7111, 69, 70ltleii 8957 . . . . . 6  |-  0  <_  2
72 1lt2 9902 . . . . . . 7  |-  1  <  2
7312, 69, 72ltleii 8957 . . . . . 6  |-  1  <_  2
7467, 68, 71, 73keephyp 3632 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2
75 df-2 9820 . . . . 5  |-  2  =  ( 1  +  1 )
7674, 75breqtri 4062 . . . 4  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
( 1  +  1 )
7776a1i 10 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  <_  (
1  +  1 ) )
78 rnggrp 15362 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
7910, 78syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
80793ad2ant1 976 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Grp )
81 eqid 2296 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
823, 81grpcl 14511 . . . . 5  |-  ( ( R  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  R ) z )  e.  B )
8380, 35, 36, 82syl3anc 1182 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( +g  `  R ) z )  e.  B
)
84 eqeq1 2302 . . . . . 6  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  =  .0.  <->  ( y
( +g  `  R ) z )  =  .0.  ) )
8584ifbid 3596 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 ) )
8619, 25ifex 3636 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  e. 
_V
8785, 15, 86fvmpt 5618 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( F `  ( y ( +g  `  R ) z ) )  =  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 ) )
8883, 87syl 15 . . 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 5892 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  +  ( F `  z
) )  =  ( 1  +  1 ) )
9077, 88, 893brtr4d 4069 . 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 15601 1  |-  ( ph  ->  F  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884   2c2 9811   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Grpcgrp 14378   Ringcrg 15353  AbsValcabv 15597
This theorem is referenced by:  abvtriv  15622  abvn0b  16059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ico 10678  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-abv 15598
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