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Theorem m1m1sr 8128
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 8100 . . 3 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
21, 1oveq12i 6097 . 2 |- ( -1R .R -1R ) = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
3 df-1r 8099 . . 3 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
4 1pr 7921 . . . . 5 |- 1P e. P.
5 addclpr 7904 . . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
64, 4, 5mp2an 430 . . . . 5 |- ( 1P +P. 1P ) e. P.
7 mulsrpr 8113 . . . . 5 |- ( ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) /\ ( 1P e. P. /\ ( 1P +P. 1P ) e. 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 431 . . . 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 7939 . . . . . . . . 9 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) e. P. )
104, 6, 9mp2an 430 . . . . . . . 8 |- ( 1P .P. ( 1P +P. 1P ) ) e. P.
11 mulclpr 7939 . . . . . . . . 9 |- ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) -> ( ( 1P +P. 1P ) .P. 1P ) e. P. )
126, 4, 11mp2an 430 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. 1P ) e. P.
13 addclpr 7904 . . . . . . . 8 |- ( ( ( 1P .P. ( 1P +P. 1P ) ) e. P. /\ ( ( 1P +P. 1P ) .P. 1P ) e. P. ) -> ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P. )
1410, 12, 13mp2an 430 . . . . . . 7 |- ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P.
15 addassprg 7946 . . . . . . 7 |- ( ( 1P e. P. /\ 1P e. P. /\ ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. 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 1378 . . . . . 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 7959 . . . . . . . . 9 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
184, 17ax-mp 5 . . . . . . . 8 |- ( 1P .P. 1P ) = 1P
19 distrprg 7955 . . . . . . . . . 10 |- ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. /\ 1P e. P. ) -> ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) )
206, 4, 4, 19mp3an 1378 . . . . . . . . 9 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
21 mulcomprg 7947 . . . . . . . . . . 11 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P ) )
224, 6, 21mp2an 430 . . . . . . . . . 10 |- ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P )
2322oveq1i 6095 . . . . . . . . 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 2262 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
2518, 24oveq12i 6097 . . . . . . 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 6096 . . . . . 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 2262 . . . . 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 7939 . . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P .P. 1P ) e. P. )
294, 4, 28mp2an 430 . . . . . . 7 |- ( 1P .P. 1P ) e. P.
30 mulclpr 7939 . . . . . . . 8 |- ( ( ( 1P +P. 1P ) e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. )
316, 6, 30mp2an 430 . . . . . . 7 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P.
32 addclpr 7904 . . . . . . 7 |- ( ( ( 1P .P. 1P ) e. P. /\ ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. ) -> ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P. )
3329, 31, 32mp2an 430 . . . . . 6 |- ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P.
34 enreceq 8103 . . . . . 6 |- ( ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) /\ ( ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P. /\ ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. 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 431 . . . . 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 146 . . . 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 2262 . . 3 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
383, 37eqtr4i 2262 . 2 |- 1R = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
392, 38eqtr4i 2262 1 |- ( -1R .R -1R ) = 1R
Colors of variables:    wff set class
This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712  (class class class)co 6085   [cec 6805   P.cnp 7658   1Pc1p 7659   +P. cpp 7660   .P. cmp 7661   ~R cer 7663   1Rc1r 7666   -1Rcm1r 7667   .R cmr 7669
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-imp 7836  df-enr 8093  df-nr 8094  df-mr 8096  df-1r 8099  df-m1r 8100
This theorem is used by: (None)
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