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Theorem prsradd 7701
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 7469 . . . 4 |- 1P e. P.
2 addclpr 7452 . . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
31, 2mpan2 422 . . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
4 addclpr 7452 . . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
51, 4mpan2 422 . . 3 |- ( B e. P. -> ( B +P. 1P ) e. P. )
6 addsrpr 7660 . . . . 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 432 . . . 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 435 . . 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 7493 . . . . . . . 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 7494 . . . . . . . 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 7452 . . . . . . . 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 6002 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( A +P. B ) +P. ( 1P +P. 1P ) ) )
20 addclpr 7452 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
21 addclpr 7452 . . . . . . . . 9 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
221, 1, 21mp2an 423 . . . . . . . 8 |- ( 1P +P. 1P ) e. P.
2322a1i 9 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. 1P ) e. P. )
24 addcomprg 7493 . . . . . . 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 409 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
2619, 25eqtrd 2190 . . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
2726oveq1d 5836 . . . 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 7494 . . . . 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 1220 . . . 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 2190 . . 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 7452 . . . . 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 7452 . . . . 5 |- ( ( ( A +P. B ) e. P. /\ 1P e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
3420, 11, 33syl2anc 409 . . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
35 enreceq 7651 . . . . . 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 435 . . . . 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 432 . . . 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 409 . . 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 2191 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 963   = wceq 1335   e. wcel 2128   <.cop 3563  (class class class)co 5821   [cec 6475   P.cnp 7206   1Pc1p 7207   +P. cpp 7208   ~R cer 7211   +R cplr 7216
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-2o 6361  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7219  df-pli 7220  df-mi 7221  df-lti 7222  df-plpq 7259  df-mpq 7260  df-enq 7262  df-nqqs 7263  df-plqqs 7264  df-mqqs 7265  df-1nqqs 7266  df-rq 7267  df-ltnqqs 7268  df-enq0 7339  df-nq0 7340  df-0nq0 7341  df-plq0 7342  df-mq0 7343  df-inp 7381  df-i1p 7382  df-iplp 7383  df-enr 7641  df-nr 7642  df-plr 7643
This theorem is referenced by:  caucvgsrlemcau  7708  caucvgsrlemgt1  7710
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