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Theorem prsradd 7024
Description: Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsradd  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. ( ( A  +P.  B )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( B  +P.  1P ) ,  1P >. ]  ~R  ) )

Proof of Theorem prsradd
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pr 6806 . . . 4  |-  1P  e.  P.
2 addclpr 6789 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
31, 2mpan2 416 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  1P )  e. 
P. )
4 addclpr 6789 . . . 4  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
51, 4mpan2 416 . . 3  |-  ( B  e.  P.  ->  ( B  +P.  1P )  e. 
P. )
6 addsrpr 6984 . . . . 5  |-  ( ( ( ( A  +P.  1P )  e.  P.  /\  1P  e.  P. )  /\  ( ( B  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( B  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P ) >. ]  ~R  )
71, 6mpanl2 426 . . . 4  |-  ( ( ( A  +P.  1P )  e.  P.  /\  (
( B  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  -> 
( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [
<. ( B  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. (
( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  )
81, 7mpanr2 429 . . 3  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( B  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. (
( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  )
93, 5, 8syl2an 283 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [
<. ( B  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. (
( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  )
10 simpl 107 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  e.  P. )
111a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  1P  e.  P. )
12 simpr 108 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  B  e.  P. )
13 addcomprg 6830 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1413adantl 271 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
15 addassprg 6831 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
1615adantl 271 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( (
f  +P.  g )  +P.  h )  =  ( f  +P.  ( g  +P.  h ) ) )
17 addclpr 6789 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
1817adantl 271 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  e.  P. )
1910, 11, 12, 14, 16, 11, 18caov4d 5716 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( A  +P.  B )  +P.  ( 1P  +P.  1P ) ) )
20 addclpr 6789 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21 addclpr 6789 . . . . . . . . 9  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
221, 1, 21mp2an 417 . . . . . . . 8  |-  ( 1P 
+P.  1P )  e.  P.
2322a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  1P )  e.  P. )
24 addcomprg 6830 . . . . . . 7  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( ( A  +P.  B )  +P.  ( 1P 
+P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  B ) ) )
2520, 23, 24syl2anc 403 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  ( 1P 
+P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  B ) ) )
2619, 25eqtrd 2114 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( 1P 
+P.  1P )  +P.  ( A  +P.  B ) ) )
2726oveq1d 5558 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( (
( 1P  +P.  1P )  +P.  ( A  +P.  B ) )  +P.  1P ) )
28 addassprg 6831 . . . . 5  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  ( A  +P.  B )  e. 
P.  /\  1P  e.  P. )  ->  ( ( ( 1P  +P.  1P )  +P.  ( A  +P.  B ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) )
2923, 20, 11, 28syl3anc 1170 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( 1P 
+P.  1P )  +P.  ( A  +P.  B ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) )
3027, 29eqtrd 2114 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) )
31 addclpr 6789 . . . . 5  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
323, 5, 31syl2an 283 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
33 addclpr 6789 . . . . 5  |-  ( ( ( A  +P.  B
)  e.  P.  /\  1P  e.  P. )  -> 
( ( A  +P.  B )  +P.  1P )  e.  P. )
3420, 11, 33syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  1P )  e.  P. )
35 enreceq 6975 . . . . . 6  |-  ( ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  /\  ( ( ( A  +P.  B )  +P. 
1P )  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P ) >. ]  ~R  =  [ <. ( ( A  +P.  B )  +P. 
1P ) ,  1P >. ]  ~R  <->  ( (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) ) )
361, 35mpanr2 429 . . . . 5  |-  ( ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  /\  ( ( A  +P.  B )  +P.  1P )  e.  P. )  -> 
( [ <. (
( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  =  [ <. ( ( A  +P.  B )  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) ) )
3722, 36mpanl2 426 . . . 4  |-  ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P.  /\  (
( A  +P.  B
)  +P.  1P )  e.  P. )  ->  ( [ <. ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  =  [ <. ( ( A  +P.  B )  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) ) )
3832, 34, 37syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( [ <. (
( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  =  [ <. ( ( A  +P.  B )  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( ( A  +P.  B )  +P.  1P ) ) ) )
3930, 38mpbird 165 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) ,  ( 1P  +P.  1P )
>. ]  ~R  =  [ <. ( ( A  +P.  B )  +P.  1P ) ,  1P >. ]  ~R  )
409, 39eqtr2d 2115 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. ( ( A  +P.  B )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( B  +P.  1P ) ,  1P >. ]  ~R  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3409  (class class class)co 5543   [cec 6170   P.cnp 6543   1Pc1p 6544    +P. cpp 6545    ~R cer 6548    +R cplr 6553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-enr 6965  df-nr 6966  df-plr 6967
This theorem is referenced by:  caucvgsrlemcau  7031  caucvgsrlemgt1  7033
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