Theorem prsradd 7898

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.

Step	Hyp	Ref	Expression
1		1pr 7666	. . . 4 |- 1P e. P.
2		addclpr 7649	. . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
3	1, 2	mpan2 425	. . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
4		addclpr 7649	. . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
5	1, 4	mpan2 425	. . 3 |- ( B e. P. -> ( B +P. 1P ) e. P. )
6		addsrpr 7857	. . . . 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 )
7	1, 6	mpanl2 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 )
8	1, 7	mpanr2 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 )
9	3, 5, 8	syl2an 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. )
11	1	a1i 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 7690	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
14	13	adantl 277	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
15		addassprg 7691	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
16	15	adantl 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 7649	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
18	17	adantl 277	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) e. P. )
19	10, 11, 12, 14, 16, 11, 18	caov4d 6130	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( A +P. B ) +P. ( 1P +P. 1P ) ) )
20		addclpr 7649	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
21		addclpr 7649	. . . . . . . . 9 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
22	1, 1, 21	mp2an 426	. . . . . . . 8 |- ( 1P +P. 1P ) e. P.
23	22	a1i 9	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. 1P ) e. P. )
24		addcomprg 7690	. . . . . . 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 ) ) )
25	20, 23, 24	syl2anc 411	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
26	19, 25	eqtrd 2237	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
27	26	oveq1d 5958	. . . 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 7691	. . . . 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 ) ) )
29	23, 20, 11, 28	syl3anc 1249	. . . 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 ) ) )
30	27, 29	eqtrd 2237	. . 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 7649	. . . . 5 |- ( ( ( A +P. 1P ) e. P. /\ ( B +P. 1P ) e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
32	3, 5, 31	syl2an 289	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
33		addclpr 7649	. . . . 5 |- ( ( ( A +P. B ) e. P. /\ 1P e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
34	20, 11, 33	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) +P. 1P ) e. P. )
35		enreceq 7848	. . . . . 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 ) ) ) )
36	1, 35	mpanr2 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 ) ) ) )
37	22, 36	mpanl2 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 ) ) ) )
38	32, 34, 37	syl2anc 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 ) ) ) )
39	30, 38	mpbird 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 )
40	9, 39	eqtr2d 2238	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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   <.cop 3635  (class class class)co 5943   [cec 6617   P.cnp 7403   1Pc1p 7404   +P. cpp 7405   ~R cer 7408   +R cplr 7413

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-i1p 7579  df-iplp 7580  df-enr 7838  df-nr 7839  df-plr 7840

This theorem is referenced by:  caucvgsrlemcau  7905  caucvgsrlemgt1  7907