Theorem pitonnlem1p1 7858

Description: Lemma for pitonn 7860. 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 7566	. . . . . 6 |- 1P e. P.
2		addclpr 7549	. . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
3	1, 1, 2	mp2an 426	. . . . 5 |- ( 1P +P. 1P ) e. P.
4		addcomprg 7590	. . . . 5 |- ( ( A e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( A +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. A ) )
5	3, 4	mpan2 425	. . . 4 |- ( A e. P. -> ( A +P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) +P. A ) )
6	5	oveq1d 5903	. . 3 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) )
7		addassprg 7591	. . . 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 1338	. . 3 |- ( A e. P. -> ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
9	6, 8	eqtrd 2220	. 2 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
10		addclpr 7549	. . . 4 |- ( ( A e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( A +P. ( 1P +P. 1P ) ) e. P. )
11	3, 10	mpan2 425	. . 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 7549	. . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
14	1, 13	mpan2 425	. . 3 |- ( A e. P. -> ( A +P. 1P ) e. P. )
15	1	a1i 9	. . 3 |- ( A e. P. -> 1P e. P. )
16		enreceq 7748	. . 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 1249	. 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 )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   e. wcel 2158   <.cop 3607  (class class class)co 5888   [cec 6546   P.cnp 7303   1Pc1p 7304   +P. cpp 7305   ~R cer 7308

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-i1p 7479  df-iplp 7480  df-enr 7738

This theorem is referenced by:  pitonnlem2  7859