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Theorem prsradd 8159
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 7927 . . . 4 |- 1P e. P.
2 addclpr 7910 . . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
31, 2mpan2 429 . . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
4 addclpr 7910 . . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
51, 4mpan2 429 . . 3 |- ( B e. P. -> ( B +P. 1P ) e. P. )
6 addsrpr 8118 . . . . 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 439 . . . 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 442 . . 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 7951 . . . . . . . 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 7952 . . . . . . . 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 7910 . . . . . . . 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 6277 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( A +P. B ) +P. ( 1P +P. 1P ) ) )
20 addclpr 7910 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
21 addclpr 7910 . . . . . . . . 9 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
221, 1, 21mp2an 430 . . . . . . . 8 |- ( 1P +P. 1P ) e. P.
2322a1i 9 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. 1P ) e. P. )
24 addcomprg 7951 . . . . . . 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 415 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
2619, 25eqtrd 2271 . . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
2726oveq1d 6103 . . . 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 7952 . . . . 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 1278 . . . 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 2271 . . 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 7910 . . . . 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 7910 . . . . 5 |- ( ( ( A +P. B ) e. P. /\ 1P e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
3420, 11, 33syl2anc 415 . . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
35 enreceq 8109 . . . . . 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 442 . . . . 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 439 . . . 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 415 . . 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 2272 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712  (class class class)co 6088   [cec 6809   P.cnp 7664   1Pc1p 7665   +P. cpp 7666   ~R cer 7669   +R cplr 7674
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-i1p 7840  df-iplp 7841  df-enr 8099  df-nr 8100  df-plr 8101
This theorem is used by:  caucvgsrlemcau  8166  caucvgsrlemgt1  8168
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