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Theorem prsradd 7562
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 7330 . . . 4  |-  1P  e.  P.
2 addclpr 7313 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
31, 2mpan2 421 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  1P )  e. 
P. )
4 addclpr 7313 . . . 4  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
51, 4mpan2 421 . . 3  |-  ( B  e.  P.  ->  ( B  +P.  1P )  e. 
P. )
6 addsrpr 7521 . . . . 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 431 . . . 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 434 . . 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 287 . 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 108 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  e.  P. )
111a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  1P  e.  P. )
12 simpr 109 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  B  e.  P. )
13 addcomprg 7354 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1413adantl 275 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
15 addassprg 7355 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
1615adantl 275 . . . . . . 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 7313 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
1817adantl 275 . . . . . . 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 5923 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( A  +P.  B )  +P.  ( 1P  +P.  1P ) ) )
20 addclpr 7313 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21 addclpr 7313 . . . . . . . . 9  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
221, 1, 21mp2an 422 . . . . . . . 8  |-  ( 1P 
+P.  1P )  e.  P.
2322a1i 9 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  1P )  e.  P. )
24 addcomprg 7354 . . . . . . 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 408 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  ( 1P 
+P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  B ) ) )
2619, 25eqtrd 2150 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  =  ( ( 1P 
+P.  1P )  +P.  ( A  +P.  B ) ) )
2726oveq1d 5757 . . . 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 7355 . . . . 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 1201 . . . 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 2150 . . 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 7313 . . . . 5  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
323, 5, 31syl2an 287 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
33 addclpr 7313 . . . . 5  |-  ( ( ( A  +P.  B
)  e.  P.  /\  1P  e.  P. )  -> 
( ( A  +P.  B )  +P.  1P )  e.  P. )
3420, 11, 33syl2anc 408 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  +P.  1P )  e.  P. )
35 enreceq 7512 . . . . . 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 434 . . . . 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 431 . . . 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 408 . . 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 166 . 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 2151 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   <.cop 3500  (class class class)co 5742   [cec 6395   P.cnp 7067   1Pc1p 7068    +P. cpp 7069    ~R cer 7072    +R cplr 7077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-enr 7502  df-nr 7503  df-plr 7504
This theorem is referenced by:  caucvgsrlemcau  7569  caucvgsrlemgt1  7571
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