Theorem prsradd 7846

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 7614	. . . 4 |- 1P e. P.
2		addclpr 7597	. . . 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 7597	. . . 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 7805	. . . . 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 7638	. . . . . . . 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 7639	. . . . . . . 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 7597	. . . . . . . 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 6103	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( A +P. B ) +P. ( 1P +P. 1P ) ) )
20		addclpr 7597	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
21		addclpr 7597	. . . . . . . . 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 7638	. . . . . . 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 2226	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) = ( ( 1P +P. 1P ) +P. ( A +P. B ) ) )
27	26	oveq1d 5933	. . . 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 7639	. . . . 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 2226	. . 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 7597	. . . . 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 7597	. . . . 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 7796	. . . . . 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 2227	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 1364   e. wcel 2164   <.cop 3621  (class class class)co 5918   [cec 6585   P.cnp 7351   1Pc1p 7352   +P. cpp 7353   ~R cer 7356   +R cplr 7361

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-enr 7786  df-nr 7787  df-plr 7788

This theorem is referenced by:  caucvgsrlemcau  7853  caucvgsrlemgt1  7855