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Theorem pitonnlem1p1 7760
 Description: Lemma for pitonn 7762. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
Assertion
Ref Expression
pitonnlem1p1 (𝐴P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )

Proof of Theorem pitonnlem1p1
StepHypRef Expression
1 1pr 7468 . . . . . 6 1PP
2 addclpr 7451 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
31, 1, 2mp2an 423 . . . . 5 (1P +P 1P) ∈ P
4 addcomprg 7492 . . . . 5 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
53, 4mpan2 422 . . . 4 (𝐴P → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
65oveq1d 5836 . . 3 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P 𝐴) +P 1P))
7 addassprg 7493 . . . 4 (((1P +P 1P) ∈ P𝐴P ∧ 1PP) → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
83, 1, 7mp3an13 1310 . . 3 (𝐴P → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
96, 8eqtrd 2190 . 2 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
10 addclpr 7451 . . . 4 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) ∈ P)
113, 10mpan2 422 . . 3 (𝐴P → (𝐴 +P (1P +P 1P)) ∈ P)
123a1i 9 . . 3 (𝐴P → (1P +P 1P) ∈ P)
13 addclpr 7451 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
141, 13mpan2 422 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
151a1i 9 . . 3 (𝐴P → 1PP)
16 enreceq 7650 . . 3 ((((𝐴 +P (1P +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((𝐴 +P 1P) ∈ P ∧ 1PP)) → ([⟨(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 1221 . 2 (𝐴P → ([⟨(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 166 1 (𝐴P → [⟨(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 104   = wceq 1335   ∈ wcel 2128  ⟨cop 3563  (class class class)co 5821  [cec 6475  Pcnp 7205  1Pc1p 7206   +P cpp 7207   ~R cer 7210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-2o 6361  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7218  df-pli 7219  df-mi 7220  df-lti 7221  df-plpq 7258  df-mpq 7259  df-enq 7261  df-nqqs 7262  df-plqqs 7263  df-mqqs 7264  df-1nqqs 7265  df-rq 7266  df-ltnqqs 7267  df-enq0 7338  df-nq0 7339  df-0nq0 7340  df-plq0 7341  df-mq0 7342  df-inp 7380  df-i1p 7381  df-iplp 7382  df-enr 7640 This theorem is referenced by:  pitonnlem2  7761
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