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Theorem m1m1sr 6649
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 6621 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 5467 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 6620 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 6534 . . . . 5 1P P
5 addclpr 6520 . . . . . 6 ((1P P 1P P) → (1P +P 1P) P)
64, 4, 5mp2an 402 . . . . 5 (1P +P 1P) P
7 mulsrpr 6634 . . . . 5 (((1P P (1P +P 1P) P) (1P P (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 6551 . . . . . . . . 9 ((1P P (1P +P 1P) P) → (1P ·P (1P +P 1P)) P)
104, 6, 9mp2an 402 . . . . . . . 8 (1P ·P (1P +P 1P)) P
11 mulclpr 6551 . . . . . . . . 9 (((1P +P 1P) P 1P P) → ((1P +P 1P) ·P 1P) P)
126, 4, 11mp2an 402 . . . . . . . 8 ((1P +P 1P) ·P 1P) P
13 addclpr 6520 . . . . . . . 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 6553 . . . . . . 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 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 1231 . . . . . 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 6566 . . . . . . . . 9 (1P P → (1P ·P 1P) = 1P)
184, 17ax-mp 7 . . . . . . . 8 (1P ·P 1P) = 1P
19 distrprg 6562 . . . . . . . . . 10 (((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)))
206, 4, 4, 19mp3an 1231 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
21 mulcomprg 6554 . . . . . . . . . . 11 ((1P P (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 5465 . . . . . . . . 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 2060 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
2518, 24oveq12i 5467 . . . . . . 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 5466 . . . . . 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 2060 . . . . 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 6551 . . . . . . . 8 ((1P P 1P P) → (1P ·P 1P) P)
294, 4, 28mp2an 402 . . . . . . 7 (1P ·P 1P) P
30 mulclpr 6551 . . . . . . . 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 6520 . . . . . . 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 6624 . . . . . 6 ((((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) ·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 2060 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
383, 37eqtr4i 2060 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
392, 38eqtr4i 2060 1 (-1R ·R -1R) = 1R
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  cop 3370  (class class class)co 5455  [cec 6040  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ·P cmp 6278   ~R cer 6280  1Rc1r 6283  -1Rcm1r 6284   ·R cmr 6286
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-imp 6451  df-enr 6614  df-nr 6615  df-mr 6617  df-1r 6620  df-m1r 6621
This theorem is referenced by: (None)
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