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Theorem pitonnlem1p1 6920
Description: Lemma for pitonn 6922. 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 6650 . . . . . 6 1PP
2 addclpr 6633 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
31, 1, 2mp2an 402 . . . . 5 (1P +P 1P) ∈ P
4 addcomprg 6674 . . . . 5 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
53, 4mpan2 401 . . . 4 (𝐴P → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
65oveq1d 5527 . . 3 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P 𝐴) +P 1P))
7 addassprg 6675 . . . 4 (((1P +P 1P) ∈ P𝐴P ∧ 1PP) → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
83, 1, 7mp3an13 1223 . . 3 (𝐴P → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
96, 8eqtrd 2072 . 2 (𝐴P → ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
10 addclpr 6633 . . . 4 ((𝐴P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) ∈ P)
113, 10mpan2 401 . . 3 (𝐴P → (𝐴 +P (1P +P 1P)) ∈ P)
123a1i 9 . . 3 (𝐴P → (1P +P 1P) ∈ P)
13 addclpr 6633 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
141, 13mpan2 401 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
151a1i 9 . . 3 (𝐴P → 1PP)
16 enreceq 6819 . . 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 1136 . 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 156 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 98   = wceq 1243  wcel 1393  cop 3378  (class class class)co 5512  [cec 6104  Pcnp 6387  1Pc1p 6388   +P cpp 6389   ~R cer 6392
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-enr 6809
This theorem is referenced by:  pitonnlem2  6921
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