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Theorem m1m1sr 7562
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 7534 . . 3 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
21, 1oveq12i 5779 . 2 |- ( -1R .R -1R ) = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
3 df-1r 7533 . . 3 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
4 1pr 7355 . . . . 5 |- 1P e. P.
5 addclpr 7338 . . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
64, 4, 5mp2an 422 . . . . 5 |- ( 1P +P. 1P ) e. P.
7 mulsrpr 7547 . . . . 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 423 . . . 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 7373 . . . . . . . . 9 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) e. P. )
104, 6, 9mp2an 422 . . . . . . . 8 |- ( 1P .P. ( 1P +P. 1P ) ) e. P.
11 mulclpr 7373 . . . . . . . . 9 |- ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) -> ( ( 1P +P. 1P ) .P. 1P ) e. P. )
126, 4, 11mp2an 422 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. 1P ) e. P.
13 addclpr 7338 . . . . . . . 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 422 . . . . . . 7 |- ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P.
15 addassprg 7380 . . . . . . 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 1315 . . . . . 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 7393 . . . . . . . . 9 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
184, 17ax-mp 5 . . . . . . . 8 |- ( 1P .P. 1P ) = 1P
19 distrprg 7389 . . . . . . . . . 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 1315 . . . . . . . . 9 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
21 mulcomprg 7381 . . . . . . . . . . 11 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P ) )
224, 6, 21mp2an 422 . . . . . . . . . 10 |- ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P )
2322oveq1i 5777 . . . . . . . . 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 2161 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
2518, 24oveq12i 5779 . . . . . . 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 5778 . . . . . 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 2161 . . . . 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 7373 . . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P .P. 1P ) e. P. )
294, 4, 28mp2an 422 . . . . . . 7 |- ( 1P .P. 1P ) e. P.
30 mulclpr 7373 . . . . . . . 8 |- ( ( ( 1P +P. 1P ) e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. )
316, 6, 30mp2an 422 . . . . . . 7 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P.
32 addclpr 7338 . . . . . . 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 422 . . . . . 6 |- ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P.
34 enreceq 7537 . . . . . 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 423 . . . . 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 145 . . . 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 2161 . . 3 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
383, 37eqtr4i 2161 . 2 |- 1R = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
392, 38eqtr4i 2161 1 |- ( -1R .R -1R ) = 1R
Colors of variables: wff set class
Syntax hints:   <-> wb 104   = wceq 1331   e. wcel 1480   <.cop 3525  (class class class)co 5767   [cec 6420   P.cnp 7092   1Pc1p 7093   +P. cpp 7094   .P. cmp 7095   ~R cer 7097   1Rc1r 7100   -1Rcm1r 7101   .R cmr 7103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-enr 7527  df-nr 7528  df-mr 7530  df-1r 7533  df-m1r 7534
This theorem is referenced by: (None)
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