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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
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 2434 . 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 9373 . . . . 5  |-  0  e.  RR
12 1re 9372 . . . . 5  |-  1  e.  RR
1311, 12keepel 3845 . . . 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 5855 . 2  |-  ( ph  ->  F : B --> RR )
173, 8rng0cl 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
2018, 15, 19fvmpt 5762 . . 3  |-  (  .0. 
e.  B  ->  ( F `  .0.  )  =  0 )
2110, 17, 203syl 20 . 2  |-  ( ph  ->  ( F `  .0.  )  =  0 )
22 0lt1 9849 . . 3  |-  0  <  1
23 eqeq1 2439 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  .0.  <->  y  =  .0.  ) )
2423ifbid 3799 . . . . . 6  |-  ( x  =  y  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( y  =  .0.  ,  0 ,  1 ) )
25 1ex 9368 . . . . . . 7  |-  1  e.  _V
2619, 25ifex 3846 . . . . . 6  |-  if ( y  =  .0.  , 
0 ,  1 )  e.  _V
2724, 15, 26fvmpt 5762 . . . . 5  |-  ( y  e.  B  ->  ( F `  y )  =  if ( y  =  .0.  ,  0 ,  1 ) )
28 ifnefalse 3789 . . . . 5  |-  ( y  =/=  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
2927, 28sylan9eq 2485 . . . 4  |-  ( ( y  e.  B  /\  y  =/=  .0.  )  -> 
( F `  y
)  =  1 )
30293adant1 999 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  ( F `  y )  =  1 )
3122, 30syl5breqr 4316 . 2  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  0  <  ( F `  y )
)
32 1t1e1 10456 . . . 4  |-  ( 1  x.  1 )  =  1
3332eqcomi 2437 . . 3  |-  1  =  ( 1  x.  1 )
34103ad2ant1 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
)
373, 6rngcl 16593 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y  .x.  z )  e.  B )
3834, 35, 36, 37syl3anc 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.  ) )
4039ifbid 3799 . . . . . 6  |-  ( x  =  ( y  .x.  z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 ) )
4119, 25ifex 3846 . . . . . 6  |-  if ( ( y  .x.  z
)  =  .0.  , 
0 ,  1 )  e.  _V
4240, 15, 41fvmpt 5762 . . . . 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 3787 . . . . 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 2465 . . 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 1008 . . . . . . 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 3787 . . . . . 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 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.  ) )
5655ifbid 3799 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5719, 25ifex 3846 . . . . . . 7  |-  if ( z  =  .0.  , 
0 ,  1 )  e.  _V
5856, 15, 57fvmpt 5762 . . . . . 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 1010 . . . . . . 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 3787 . . . . . 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 2465 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  1 )
6554, 64oveq12d 6098 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( 1  x.  1 ) )
6633, 48, 653eqtr4a 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
7167, 68, 69, 70keephyp 3842 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2
72 df-2 10367 . . . . 5  |-  2  =  ( 1  +  1 )
7371, 72breqtri 4303 . . . 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 rnggrp 16585 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
7610, 75syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
77763ad2ant1 1002 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Grp )
78 eqid 2433 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
793, 78grpcl 15530 . . . . 5  |-  ( ( R  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  R ) z )  e.  B )
8077, 35, 36, 79syl3anc 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.  ) )
8281ifbid 3799 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 ) )
8319, 25ifex 3846 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  e. 
_V
8482, 15, 83fvmpt 5762 . . . 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 6098 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  +  ( F `  z
) )  =  ( 1  +  1 ) )
8774, 85, 863brtr4d 4310 . 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 16828 1  |-  ( ph  ->  F  e.  A )
Colors of variables: wff setvar class
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
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