Theorem pitonnlem1p1 8213

Description: Lemma for pitonn 8215. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)

Assertion

Ref	Expression
pitonnlem1p1	|- ( A e. P. -> [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( A +P. 1P ) , 1P >. ] ~R )

Proof of Theorem pitonnlem1p1

Step	Hyp	Ref	Expression
1		1pr 7921	. . . . . 6 |- 1P e. P.
2		addclpr 7904	. . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
3	1, 1, 2	mp2an 430	. . . . 5 |- ( 1P +P. 1P ) e. P.
4		addcomprg 7945	. . . . 5 |- ( ( A e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( A +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. A ) )
5	3, 4	mpan2 429	. . . 4 |- ( A e. P. -> ( A +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. A ) )
6	5	oveq1d 6100	. . 3 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) )
7		addassprg 7946	. . . 4 |- ( ( ( 1P +P. 1P ) e. P. /\ A e. P. /\ 1P e. P. ) -> ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
8	3, 1, 7	mp3an13 1369	. . 3 |- ( A e. P. -> ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
9	6, 8	eqtrd 2271	. 2 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
10		addclpr 7904	. . . 4 |- ( ( A e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( A +P. ( 1P +P. 1P ) ) e. P. )
11	3, 10	mpan2 429	. . 3 |- ( A e. P. -> ( A +P. ( 1P +P. 1P ) ) e. P. )
12	3	a1i 9	. . 3 |- ( A e. P. -> ( 1P +P. 1P ) e. P. )
13		addclpr 7904	. . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
14	1, 13	mpan2 429	. . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
15	1	a1i 9	. . 3 |- ( A e. P. -> 1P e. P. )
16		enreceq 8103	. . 3 |- ( ( ( ( A +P. ( 1P +P. 1P ) ) e. P. /\ ( 1P +P. 1P ) e. P. ) /\ ( ( A +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( A +P. 1P ) , 1P >. ] ~R <-> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) ) )
17	11, 12, 14, 15, 16	syl22anc 1279	. 2 |- ( A e. P. -> ( [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( A +P. 1P ) , 1P >. ] ~R <-> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) ) )
18	9, 17	mpbird 167	1 |- ( A e. P. -> [ <. ( A +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( A +P. 1P ) , 1P >. ] ~R )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712  (class class class)co 6085   [cec 6805   P.cnp 7658   1Pc1p 7659   +P. cpp 7660   ~R cer 7663

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-enr 8093

This theorem is used by:  pitonnlem2  8214