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

Theorem m1m1sr 7576
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 7548 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21, 1oveq12i 5786 . 2  |-  ( -1R 
.R  -1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
3 df-1r 7547 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
4 1pr 7369 . . . . 5  |-  1P  e.  P.
5 addclpr 7352 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
64, 4, 5mp2an 422 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
7 mulsrpr 7561 . . . . 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 7387 . . . . . . . . 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 7387 . . . . . . . . 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 7352 . . . . . . . 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 7394 . . . . . . 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 7407 . . . . . . . . 9  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
184, 17ax-mp 5 . . . . . . . 8  |-  ( 1P 
.P.  1P )  =  1P
19 distrprg 7403 . . . . . . . . . 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 7395 . . . . . . . . . . 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 5784 . . . . . . . . 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 2163 . . . . . . . 8  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( 1P  .P.  ( 1P  +P.  1P ) )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
2518, 24oveq12i 5786 . . . . . . 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 5785 . . . . . 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 2163 . . . . 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 7387 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  .P.  1P )  e.  P. )
294, 4, 28mp2an 422 . . . . . . 7  |-  ( 1P 
.P.  1P )  e.  P.
30 mulclpr 7387 . . . . . . . 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 7352 . . . . . . 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 7551 . . . . . 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 2163 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  .R 
[ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
383, 37eqtr4i 2163 . 2  |-  1R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
392, 38eqtr4i 2163 1  |-  ( -1R 
.R  -1R )  =  1R
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    e. wcel 1480   <.cop 3530  (class class class)co 5774   [cec 6427   P.cnp 7106   1Pc1p 7107    +P. cpp 7108    .P. cmp 7109    ~R cer 7111   1Rc1r 7114   -1Rcm1r 7115    .R cmr 7117
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-i1p 7282  df-iplp 7283  df-imp 7284  df-enr 7541  df-nr 7542  df-mr 7544  df-1r 7547  df-m1r 7548
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator