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Theorem pitonnlem1p1 6979
Description: Lemma for pitonn 6981. 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 6709 . . . . . 6 1PP
2 addclpr 6692 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
31, 1, 2mp2an 410 . . . . 5 (1P +P 1P) ∈ P
4 addcomprg 6733 . . . . 5 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
53, 4mpan2 409 . . . 4 (𝐴P → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
65oveq1d 5554 . . 3 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P 𝐴) +P 1P))
7 addassprg 6734 . . . 4 (((1P +P 1P) ∈ P𝐴P ∧ 1PP) → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
83, 1, 7mp3an13 1234 . . 3 (𝐴P → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
96, 8eqtrd 2088 . 2 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
10 addclpr 6692 . . . 4 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) ∈ P)
113, 10mpan2 409 . . 3 (𝐴P → (𝐴 +P (1P +P 1P)) ∈ P)
123a1i 9 . . 3 (𝐴P → (1P +P 1P) ∈ P)
13 addclpr 6692 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
141, 13mpan2 409 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
151a1i 9 . . 3 (𝐴P → 1PP)
16 enreceq 6878 . . 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 1147 . 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 160 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 102   = wceq 1259  wcel 1409  cop 3405  (class class class)co 5539  [cec 6134  Pcnp 6446  1Pc1p 6447   +P cpp 6448   ~R cer 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-enr 6868
This theorem is referenced by:  pitonnlem2  6980
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