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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
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 2468 . 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 9608 . . . . 5  |-  0  e.  RR
12 1re 9607 . . . . 5  |-  1  e.  RR
1311, 12keepel 4013 . . . 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 6056 . 2  |-  ( ph  ->  F : B --> RR )
173, 8ring0cl 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
2018, 15, 19fvmpt 5957 . . 3  |-  (  .0. 
e.  B  ->  ( F `  .0.  )  =  0 )
2110, 17, 203syl 20 . 2  |-  ( ph  ->  ( F `  .0.  )  =  0 )
22 0lt1 10087 . . 3  |-  0  <  1
23 eqeq1 2471 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  .0.  <->  y  =  .0.  ) )
2423ifbid 3967 . . . . . 6  |-  ( x  =  y  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( y  =  .0.  ,  0 ,  1 ) )
25 1ex 9603 . . . . . . 7  |-  1  e.  _V
2619, 25ifex 4014 . . . . . 6  |-  if ( y  =  .0.  , 
0 ,  1 )  e.  _V
2724, 15, 26fvmpt 5957 . . . . 5  |-  ( y  e.  B  ->  ( F `  y )  =  if ( y  =  .0.  ,  0 ,  1 ) )
28 ifnefalse 3957 . . . . 5  |-  ( y  =/=  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
2927, 28sylan9eq 2528 . . . 4  |-  ( ( y  e.  B  /\  y  =/=  .0.  )  -> 
( F `  y
)  =  1 )
30293adant1 1014 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  ( F `  y )  =  1 )
3122, 30syl5breqr 4489 . 2  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  0  <  ( F `  y )
)
32 1t1e1 10695 . . . 4  |-  ( 1  x.  1 )  =  1
3332eqcomi 2480 . . 3  |-  1  =  ( 1  x.  1 )
34103ad2ant1 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
)
373, 6ringcl 17084 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y  .x.  z )  e.  B )
3834, 35, 36, 37syl3anc 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.  ) )
4039ifbid 3967 . . . . . 6  |-  ( x  =  ( y  .x.  z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 ) )
4119, 25ifex 4014 . . . . . 6  |-  if ( ( y  .x.  z
)  =  .0.  , 
0 ,  1 )  e.  _V
4240, 15, 41fvmpt 5957 . . . . 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 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 )
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 2508 . . 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 1023 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  =/=  .0.  )
5150neneqd 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 )
5351, 52syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( y  =  .0.  ,  0 ,  1 )  =  1 )
5449, 53eqtrd 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.  ) )
5655ifbid 3967 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5719, 25ifex 4014 . . . . . . 7  |-  if ( z  =  .0.  , 
0 ,  1 )  e.  _V
5856, 15, 57fvmpt 5957 . . . . . 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 1025 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  =/=  .0.  )
6160neneqd 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 )
6361, 62syl 16 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( z  =  .0.  ,  0 ,  1 )  =  1 )
6459, 63eqtrd 2508 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  1 )
6554, 64oveq12d 6313 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( 1  x.  1 ) )
6633, 48, 653eqtr4a 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
7167, 68, 69, 70keephyp 4010 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2
72 df-2 10606 . . . . 5  |-  2  =  ( 1  +  1 )
7371, 72breqtri 4476 . . . 4  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
( 1  +  1 )
7473a1i 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 )
7610, 75syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
77763ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Grp )
78 eqid 2467 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
793, 78grpcl 15935 . . . . 5  |-  ( ( R  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  R ) z )  e.  B )
8077, 35, 36, 79syl3anc 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.  ) )
8281ifbid 3967 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 ) )
8319, 25ifex 4014 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  e. 
_V
8482, 15, 83fvmpt 5957 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( F `  ( y ( +g  `  R ) z ) )  =  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 ) )
8580, 84syl 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 ) )
8654, 64oveq12d 6313 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  +  ( F `  z
) )  =  ( 1  +  1 ) )
8774, 85, 863brtr4d 4483 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y ( +g  `  R ) z ) )  <_  ( ( F `  y )  +  ( F `  z ) ) )
882, 4, 5, 7, 9, 10, 16, 21, 31, 66, 87isabvd 17340 1  |-  ( ph  ->  F  e.  A )
Colors of variables: wff setvar class
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
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