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Theorem prsradd 7785
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 7553 . . . 4  |-  1P  e.  P.
2 addclpr 7536 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
31, 2mpan2 425 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  1P )  e. 
P. )
4 addclpr 7536 . . . 4  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
51, 4mpan2 425 . . 3  |-  ( B  e.  P.  ->  ( B  +P.  1P )  e. 
P. )
6 addsrpr 7744 . . . . 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 435 . . . 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 438 . . 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 289 . 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 109 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  e.  P. )
111a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  1P  e.  P. )
12 simpr 110 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  B  e.  P. )
13 addcomprg 7577 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1413adantl 277 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
15 addassprg 7578 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
1615adantl 277 . . . . . . 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 7536 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
1817adantl 277 . . . . . . 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 6059 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( A  +P.  B )  +P.  ( 1P  +P.  1P ) ) )
20 addclpr 7536 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21 addclpr 7536 . . . . . . . . 9  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
221, 1, 21mp2an 426 . . . . . . . 8  |-  ( 1P 
+P.  1P )  e.  P.
2322a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  1P )  e.  P. )
24 addcomprg 7577 . . . . . . 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 411 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  ( 1P 
+P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  B ) ) )
2619, 25eqtrd 2210 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( 1P 
+P.  1P )  +P.  ( A  +P.  B ) ) )
2726oveq1d 5890 . . . 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 7578 . . . . 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 1238 . . . 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 2210 . . 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 7536 . . . . 5  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
323, 5, 31syl2an 289 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
33 addclpr 7536 . . . . 5  |-  ( ( ( A  +P.  B
)  e.  P.  /\  1P  e.  P. )  -> 
( ( A  +P.  B )  +P.  1P )  e.  P. )
3420, 11, 33syl2anc 411 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  1P )  e.  P. )
35 enreceq 7735 . . . . . 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 438 . . . . 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 435 . . . 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 411 . . 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 167 . 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 2211 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   <.cop 3596  (class class class)co 5875   [cec 6533   P.cnp 7290   1Pc1p 7291    +P. cpp 7292    ~R cer 7295    +R cplr 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-enr 7725  df-nr 7726  df-plr 7727
This theorem is referenced by:  caucvgsrlemcau  7792  caucvgsrlemgt1  7794
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