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Theorem m1m1sr 7735
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 7707 . . 3 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
21, 1oveq12i 5877 . 2 |- ( -1R .R -1R ) = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
3 df-1r 7706 . . 3 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
4 1pr 7528 . . . . 5 |- 1P e. P.
5 addclpr 7511 . . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
64, 4, 5mp2an 426 . . . . 5 |- ( 1P +P. 1P ) e. P.
7 mulsrpr 7720 . . . . 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 427 . . . 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 7546 . . . . . . . . 9 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) e. P. )
104, 6, 9mp2an 426 . . . . . . . 8 |- ( 1P .P. ( 1P +P. 1P ) ) e. P.
11 mulclpr 7546 . . . . . . . . 9 |- ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) -> ( ( 1P +P. 1P ) .P. 1P ) e. P. )
126, 4, 11mp2an 426 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. 1P ) e. P.
13 addclpr 7511 . . . . . . . 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 426 . . . . . . 7 |- ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P.
15 addassprg 7553 . . . . . . 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 1337 . . . . . 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 7566 . . . . . . . . 9 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
184, 17ax-mp 5 . . . . . . . 8 |- ( 1P .P. 1P ) = 1P
19 distrprg 7562 . . . . . . . . . 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 1337 . . . . . . . . 9 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
21 mulcomprg 7554 . . . . . . . . . . 11 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P ) )
224, 6, 21mp2an 426 . . . . . . . . . 10 |- ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P )
2322oveq1i 5875 . . . . . . . . 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 2199 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
2518, 24oveq12i 5877 . . . . . . 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 5876 . . . . . 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 2199 . . . . 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 7546 . . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P .P. 1P ) e. P. )
294, 4, 28mp2an 426 . . . . . . 7 |- ( 1P .P. 1P ) e. P.
30 mulclpr 7546 . . . . . . . 8 |- ( ( ( 1P +P. 1P ) e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. )
316, 6, 30mp2an 426 . . . . . . 7 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P.
32 addclpr 7511 . . . . . . 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 426 . . . . . 6 |- ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P.
34 enreceq 7710 . . . . . 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 427 . . . . 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 2199 . . 3 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
383, 37eqtr4i 2199 . 2 |- 1R = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
392, 38eqtr4i 2199 1 |- ( -1R .R -1R ) = 1R
Colors of variables: wff set class
Syntax hints:   <-> wb 105   = wceq 1353   e. wcel 2146   <.cop 3592  (class class class)co 5865   [cec 6523   P.cnp 7265   1Pc1p 7266   +P. cpp 7267   .P. cmp 7268   ~R cer 7270   1Rc1r 7273   -1Rcm1r 7274   .R cmr 7276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-i1p 7441  df-iplp 7442  df-imp 7443  df-enr 7700  df-nr 7701  df-mr 7703  df-1r 7706  df-m1r 7707
This theorem is referenced by: (None)
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