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Theorem abvtrivd 18068
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 2452 . 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 9643 . . . . 5  |-  0  e.  RR
12 1re 9642 . . . . 5  |-  1  e.  RR
1311, 12keepel 3948 . . . 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 6046 . 2  |-  ( ph  ->  F : B --> RR )
173, 8ring0cl 17802 . . 3  |-  ( R  e.  Ring  ->  .0.  e.  B )
18 iftrue 3887 . . . 4  |-  ( x  =  .0.  ->  if ( x  =  .0.  ,  0 ,  1 )  =  0 )
19 c0ex 9637 . . . 4  |-  0  e.  _V
2018, 15, 19fvmpt 5948 . . 3  |-  (  .0. 
e.  B  ->  ( F `  .0.  )  =  0 )
2110, 17, 203syl 18 . 2  |-  ( ph  ->  ( F `  .0.  )  =  0 )
22 0lt1 10136 . . 3  |-  0  <  1
23 eqeq1 2455 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  .0.  <->  y  =  .0.  ) )
2423ifbid 3903 . . . . . 6  |-  ( x  =  y  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( y  =  .0.  ,  0 ,  1 ) )
25 1ex 9638 . . . . . . 7  |-  1  e.  _V
2619, 25ifex 3949 . . . . . 6  |-  if ( y  =  .0.  , 
0 ,  1 )  e.  _V
2724, 15, 26fvmpt 5948 . . . . 5  |-  ( y  e.  B  ->  ( F `  y )  =  if ( y  =  .0.  ,  0 ,  1 ) )
28 ifnefalse 3893 . . . . 5  |-  ( y  =/=  .0.  ->  if ( y  =  .0. 
,  0 ,  1 )  =  1 )
2927, 28sylan9eq 2505 . . . 4  |-  ( ( y  e.  B  /\  y  =/=  .0.  )  -> 
( F `  y
)  =  1 )
30293adant1 1026 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  ( F `  y )  =  1 )
3122, 30syl5breqr 4439 . 2  |-  ( (
ph  /\  y  e.  B  /\  y  =/=  .0.  )  ->  0  <  ( F `  y )
)
32 1t1e1 10757 . . . 4  |-  ( 1  x.  1 )  =  1
3332eqcomi 2460 . . 3  |-  1  =  ( 1  x.  1 )
34103ad2ant1 1029 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Ring )
35 simp2l 1034 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  e.  B
)
36 simp3l 1036 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  e.  B
)
373, 6ringcl 17794 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y  .x.  z )  e.  B )
3834, 35, 36, 37syl3anc 1268 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y  .x.  z )  e.  B
)
39 eqeq1 2455 . . . . . . 7  |-  ( x  =  ( y  .x.  z )  ->  (
x  =  .0.  <->  ( y  .x.  z )  =  .0.  ) )
4039ifbid 3903 . . . . . 6  |-  ( x  =  ( y  .x.  z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 ) )
4119, 25ifex 3949 . . . . . 6  |-  if ( ( y  .x.  z
)  =  .0.  , 
0 ,  1 )  e.  _V
4240, 15, 41fvmpt 5948 . . . . 5  |-  ( ( y  .x.  z )  e.  B  ->  ( F `  ( y  .x.  z ) )  =  if ( ( y 
.x.  z )  =  .0.  ,  0 ,  1 ) )
4338, 42syl 17 . . . 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 2629 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  ( y  .x.  z )  =  .0.  )
4645iffalsed 3892 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y  .x.  z )  =  .0.  ,  0 ,  1 )  =  1 )
4743, 46eqtrd 2485 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  1 )
4835, 27syl 17 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  if ( y  =  .0. 
,  0 ,  1 ) )
49 simp2r 1035 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  y  =/=  .0.  )
5049neneqd 2629 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  y  =  .0.  )
5150iffalsed 3892 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( y  =  .0.  ,  0 ,  1 )  =  1 )
5248, 51eqtrd 2485 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  y )  =  1 )
53 eqeq1 2455 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  .0.  <->  z  =  .0.  ) )
5453ifbid 3903 . . . . . . 7  |-  ( x  =  z  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5519, 25ifex 3949 . . . . . . 7  |-  if ( z  =  .0.  , 
0 ,  1 )  e.  _V
5654, 15, 55fvmpt 5948 . . . . . 6  |-  ( z  e.  B  ->  ( F `  z )  =  if ( z  =  .0.  ,  0 ,  1 ) )
5736, 56syl 17 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  if ( z  =  .0. 
,  0 ,  1 ) )
58 simp3r 1037 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  z  =/=  .0.  )
5958neneqd 2629 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  -.  z  =  .0.  )
6059iffalsed 3892 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( z  =  .0.  ,  0 ,  1 )  =  1 )
6157, 60eqtrd 2485 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  z )  =  1 )
6252, 61oveq12d 6308 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( 1  x.  1 ) )
6333, 47, 623eqtr4a 2511 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y  .x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
64 breq1 4405 . . . . . 6  |-  ( 0  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
0  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
65 breq1 4405 . . . . . 6  |-  ( 1  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  ->  (
1  <_  2  <->  if (
( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2 ) )
66 0le2 10700 . . . . . 6  |-  0  <_  2
67 1le2 10823 . . . . . 6  |-  1  <_  2
6864, 65, 66, 67keephyp 3945 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
2
69 df-2 10668 . . . . 5  |-  2  =  ( 1  +  1 )
7068, 69breqtri 4426 . . . 4  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  <_ 
( 1  +  1 )
7170a1i 11 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 )  <_  (
1  +  1 ) )
72 ringgrp 17785 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
7310, 72syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
74733ad2ant1 1029 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  R  e.  Grp )
75 eqid 2451 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
763, 75grpcl 16679 . . . . 5  |-  ( ( R  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  R ) z )  e.  B )
7774, 35, 36, 76syl3anc 1268 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( +g  `  R ) z )  e.  B
)
78 eqeq1 2455 . . . . . 6  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  =  .0.  <->  ( y
( +g  `  R ) z )  =  .0.  ) )
7978ifbid 3903 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  if ( x  =  .0.  ,  0 ,  1 )  =  if ( ( y ( +g  `  R
) z )  =  .0.  ,  0 ,  1 ) )
8019, 25ifex 3949 . . . . 5  |-  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 )  e. 
_V
8179, 15, 80fvmpt 5948 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( F `  ( y ( +g  `  R ) z ) )  =  if ( ( y ( +g  `  R ) z )  =  .0.  ,  0 ,  1 ) )
8277, 81syl 17 . . 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 ) )
8352, 61oveq12d 6308 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( ( F `
 y )  +  ( F `  z
) )  =  ( 1  +  1 ) )
8471, 82, 833brtr4d 4433 . 2  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( F `  ( y ( +g  `  R ) z ) )  <_  ( ( F `  y )  +  ( F `  z ) ) )
852, 4, 5, 7, 9, 10, 16, 21, 31, 63, 84isabvd 18048 1  |-  ( ph  ->  F  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676   2c2 10659   Basecbs 15121   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   Ringcrg 17780  AbsValcabv 18044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ico 11641  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ring 17782  df-abv 18045
This theorem is referenced by:  abvtriv  18069  abvn0b  18526
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