ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  m1m1sr GIF version

Theorem m1m1sr 6844
Description: Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
Assertion
Ref Expression
m1m1sr (-1R ·R -1R) = 1R

Proof of Theorem m1m1sr
StepHypRef Expression
1 df-m1r 6816 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 5524 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 6815 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 6650 . . . . 5 1PP
5 addclpr 6633 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
64, 4, 5mp2an 402 . . . . 5 (1P +P 1P) ∈ P
7 mulsrpr 6829 . . . . 5 (((1PP ∧ (1P +P 1P) ∈ P) ∧ (1PP ∧ (1P +P 1P) ∈ P)) → ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R )
84, 6, 4, 6, 7mp4an 403 . . . 4 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R
9 mulclpr 6668 . . . . . . . . 9 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
104, 6, 9mp2an 402 . . . . . . . 8 (1P ·P (1P +P 1P)) ∈ P
11 mulclpr 6668 . . . . . . . . 9 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) ·P 1P) ∈ P)
126, 4, 11mp2an 402 . . . . . . . 8 ((1P +P 1P) ·P 1P) ∈ P
13 addclpr 6633 . . . . . . . 8 (((1P ·P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) ·P 1P) ∈ P) → ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P)
1410, 12, 13mp2an 402 . . . . . . 7 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
15 addassprg 6675 . . . . . . 7 ((1PP ∧ 1PP ∧ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P) → ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)))))
164, 4, 14, 15mp3an 1232 . . . . . 6 ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))))
17 1idpr 6688 . . . . . . . . 9 (1PP → (1P ·P 1P) = 1P)
184, 17ax-mp 7 . . . . . . . 8 (1P ·P 1P) = 1P
19 distrprg 6684 . . . . . . . . . 10 (((1P +P 1P) ∈ P ∧ 1PP ∧ 1PP) → ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P)))
206, 4, 4, 19mp3an 1232 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
21 mulcomprg 6676 . . . . . . . . . . 11 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P))
224, 6, 21mp2an 402 . . . . . . . . . 10 (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
2322oveq1i 5522 . . . . . . . . 9 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
2420, 23eqtr4i 2063 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
2518, 24oveq12i 5524 . . . . . . 7 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) = (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)))
2625oveq2i 5523 . . . . . 6 (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P)))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))))
2716, 26eqtr4i 2063 . . . . 5 ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))))
28 mulclpr 6668 . . . . . . . 8 ((1PP ∧ 1PP) → (1P ·P 1P) ∈ P)
294, 4, 28mp2an 402 . . . . . . 7 (1P ·P 1P) ∈ P
30 mulclpr 6668 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
316, 6, 30mp2an 402 . . . . . . 7 ((1P +P 1P) ·P (1P +P 1P)) ∈ P
32 addclpr 6633 . . . . . . 7 (((1P ·P 1P) ∈ P ∧ ((1P +P 1P) ·P (1P +P 1P)) ∈ P) → ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P)
3329, 31, 32mp2an 402 . . . . . 6 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
34 enreceq 6819 . . . . . 6 ((((1P +P 1P) ∈ P ∧ 1PP) ∧ (((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P ∧ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P)) → ([⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R ↔ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))))))
356, 4, 33, 14, 34mp4an 403 . . . . 5 ([⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R ↔ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P)))))
3627, 35mpbir 134 . . . 4 [⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R
378, 36eqtr4i 2063 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
383, 37eqtr4i 2063 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
392, 38eqtr4i 2063 1 (-1R ·R -1R) = 1R
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1243  wcel 1393  cop 3378  (class class class)co 5512  [cec 6104  Pcnp 6387  1Pc1p 6388   +P cpp 6389   ·P cmp 6390   ~R cer 6392  1Rc1r 6395  -1Rcm1r 6396   ·R cmr 6398
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-imp 6565  df-enr 6809  df-nr 6810  df-mr 6812  df-1r 6815  df-m1r 6816
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator