Theorem pitonnlem1p1 8126

Description: Lemma for pitonn 8128. 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 7834	. . . . . 6 |- 1P e. P.
2		addclpr 7817	. . . . . 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 7858	. . . . 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 6043	. . 3 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) )
7		addassprg 7859	. . . 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 1365	. . 3 |- ( A e. P. -> ( ( ( 1P +P. 1P ) +P. A ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
9	6, 8	eqtrd 2264	. 2 |- ( A e. P. -> ( ( A +P. ( 1P +P. 1P ) ) +P. 1P ) = ( ( 1P +P. 1P ) +P. ( A +P. 1P ) ) )
10		addclpr 7817	. . . 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 7817	. . . 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 8016	. . 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 1275	. 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 1398   e. wcel 2202   <.cop 3676  (class class class)co 6028   [cec 6743   P.cnp 7571   1Pc1p 7572   +P. cpp 7573   ~R cer 7576

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-iplp 7748  df-enr 8006

This theorem is referenced by:  pitonnlem2  8127