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Theorem pitonnlem1p1 6702
Description: Lemma for pitonn 6704. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
Assertion
Ref Expression
pitonnlem1p1 (A P → [⟨(A +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(A +P 1P), 1P⟩] ~R )

Proof of Theorem pitonnlem1p1
StepHypRef Expression
1 1pr 6534 . . . . . 6 1P P
2 addclpr 6520 . . . . . 6 ((1P P 1P P) → (1P +P 1P) P)
31, 1, 2mp2an 402 . . . . 5 (1P +P 1P) P
4 addcomprg 6552 . . . . 5 ((A P (1P +P 1P) P) → (A +P (1P +P 1P)) = ((1P +P 1P) +P A))
53, 4mpan2 401 . . . 4 (A P → (A +P (1P +P 1P)) = ((1P +P 1P) +P A))
65oveq1d 5470 . . 3 (A P → ((A +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P A) +P 1P))
7 addassprg 6553 . . . 4 (((1P +P 1P) P A P 1P P) → (((1P +P 1P) +P A) +P 1P) = ((1P +P 1P) +P (A +P 1P)))
83, 1, 7mp3an13 1222 . . 3 (A P → (((1P +P 1P) +P A) +P 1P) = ((1P +P 1P) +P (A +P 1P)))
96, 8eqtrd 2069 . 2 (A P → ((A +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (A +P 1P)))
10 addclpr 6520 . . . 4 ((A P (1P +P 1P) P) → (A +P (1P +P 1P)) P)
113, 10mpan2 401 . . 3 (A P → (A +P (1P +P 1P)) P)
123a1i 9 . . 3 (A P → (1P +P 1P) P)
13 addclpr 6520 . . . 4 ((A P 1P P) → (A +P 1P) P)
141, 13mpan2 401 . . 3 (A P → (A +P 1P) P)
151a1i 9 . . 3 (A P → 1P P)
16 enreceq 6624 . . 3 ((((A +P (1P +P 1P)) P (1P +P 1P) P) ((A +P 1P) P 1P 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))))
1711, 12, 14, 15, 16syl22anc 1135 . 2 (A 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))))
189, 17mpbird 156 1 (A P → [⟨(A +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(A +P 1P), 1P⟩] ~R )
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  cop 3370  (class class class)co 5455  [cec 6040  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ~R cer 6280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-enr 6614
This theorem is referenced by:  pitonnlem2  6703
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