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Theorem prsradd 7752
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 7520 . . . 4 |- 1P e. P.
2 addclpr 7503 . . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
31, 2mpan2 423 . . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
4 addclpr 7503 . . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
51, 4mpan2 423 . . 3 |- ( B e. P. -> ( B +P. 1P ) e. P. )
6 addsrpr 7711 . . . . 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 433 . . . 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 436 . . 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 7544 . . . . . . . 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 7545 . . . . . . . 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 7503 . . . . . . . 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 6041 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( A +P. B ) +P. ( 1P +P. 1P ) ) )
20 addclpr 7503 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
21 addclpr 7503 . . . . . . . . 9 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
221, 1, 21mp2an 424 . . . . . . . 8 |- ( 1P +P. 1P ) e. P.
2322a1i 9 . . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. 1P ) e. P. )
24 addcomprg 7544 . . . . . . 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 2204 . . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
2726oveq1d 5872 . . . 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 7545 . . . . 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 1234 . . . 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 2204 . . 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 7503 . . . . 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 7503 . . . . 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 7702 . . . . . 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 436 . . . . 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 433 . . . 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 2205 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 974   = wceq 1349   e. wcel 2142   <.cop 3587  (class class class)co 5857   [cec 6515   P.cnp 7257   1Pc1p 7258   +P. cpp 7259   ~R cer 7262   +R cplr 7267
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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573
This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-eprel 4275  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-irdg 6353  df-1o 6399  df-2o 6400  df-oadd 6403  df-omul 6404  df-er 6517  df-ec 6519  df-qs 6523  df-ni 7270  df-pli 7271  df-mi 7272  df-lti 7273  df-plpq 7310  df-mpq 7311  df-enq 7313  df-nqqs 7314  df-plqqs 7315  df-mqqs 7316  df-1nqqs 7317  df-rq 7318  df-ltnqqs 7319  df-enq0 7390  df-nq0 7391  df-0nq0 7392  df-plq0 7393  df-mq0 7394  df-inp 7432  df-i1p 7433  df-iplp 7434  df-enr 7692  df-nr 7693  df-plr 7694
This theorem is referenced by:  caucvgsrlemcau  7759  caucvgsrlemgt1  7761
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