MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1m1sr Structured version   Unicode version

Theorem m1m1sr 9361
Description: Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
m1m1sr  |-  ( -1R 
.R  -1R )  =  1R

Proof of Theorem m1m1sr
StepHypRef Expression
1 df-m1r 9334 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21, 1oveq12i 6202 . 2  |-  ( -1R 
.R  -1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
3 df-1r 9333 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
4 1pr 9285 . . . . 5  |-  1P  e.  P.
5 addclpr 9288 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
64, 4, 5mp2an 672 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
7 mulsrpr 9344 . . . . 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 673 . . . 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 addasspr 9292 . . . . . 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 ) ) ) )
10 1idpr 9299 . . . . . . . . 9  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
114, 10ax-mp 5 . . . . . . . 8  |-  ( 1P 
.P.  1P )  =  1P
12 distrpr 9298 . . . . . . . . 9  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( ( 1P  +P.  1P )  .P.  1P )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
13 mulcompr 9293 . . . . . . . . . 10  |-  ( 1P 
.P.  ( 1P  +P.  1P ) )  =  ( ( 1P  +P.  1P )  .P.  1P )
1413oveq1i 6200 . . . . . . . . 9  |-  ( ( 1P  .P.  ( 1P 
+P.  1P ) )  +P.  ( ( 1P  +P.  1P )  .P.  1P ) )  =  ( ( ( 1P  +P.  1P )  .P.  1P )  +P.  ( ( 1P  +P.  1P )  .P.  1P ) )
1512, 14eqtr4i 2483 . . . . . . . 8  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( 1P  .P.  ( 1P  +P.  1P ) )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
1611, 15oveq12i 6202 . . . . . . 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 ) ) )
1716oveq2i 6201 . . . . . 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 ) ) ) )
189, 17eqtr4i 2483 . . . . 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 ) ) ) )
19 mulclpr 9290 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  .P.  1P )  e.  P. )
204, 4, 19mp2an 672 . . . . . . 7  |-  ( 1P 
.P.  1P )  e.  P.
21 mulclpr 9290 . . . . . . . 8  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  ( 1P  +P.  1P )  e. 
P. )  ->  (
( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  e.  P. )
226, 6, 21mp2an 672 . . . . . . 7  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  e.  P.
23 addclpr 9288 . . . . . . 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. )
2420, 22, 23mp2an 672 . . . . . 6  |-  ( ( 1P  .P.  1P )  +P.  ( ( 1P 
+P.  1P )  .P.  ( 1P  +P.  1P ) ) )  e.  P.
25 mulclpr 9290 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( 1P  .P.  ( 1P  +P.  1P ) )  e.  P. )
264, 6, 25mp2an 672 . . . . . . 7  |-  ( 1P 
.P.  ( 1P  +P.  1P ) )  e.  P.
27 mulclpr 9290 . . . . . . . 8  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  1P )  .P.  1P )  e. 
P. )
286, 4, 27mp2an 672 . . . . . . 7  |-  ( ( 1P  +P.  1P )  .P.  1P )  e. 
P.
29 addclpr 9288 . . . . . . 7  |-  ( ( ( 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. )
3026, 28, 29mp2an 672 . . . . . 6  |-  ( ( 1P  .P.  ( 1P 
+P.  1P ) )  +P.  ( ( 1P  +P.  1P )  .P.  1P ) )  e.  P.
31 enreceq 9337 . . . . . 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 ) ) ) ) ) )
326, 4, 24, 30, 31mp4an 673 . . . . 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 ) ) ) ) )
3318, 32mpbir 209 . . . 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
348, 33eqtr4i 2483 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  .R 
[ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
353, 34eqtr4i 2483 . 2  |-  1R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
362, 35eqtr4i 2483 1  |-  ( -1R 
.R  -1R )  =  1R
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   <.cop 3981  (class class class)co 6190   [cec 7199   P.cnp 9127   1Pc1p 9128    +P. cpp 9129    .P. cmp 9130    ~R cer 9134   1Rc1r 9137   -1Rcm1r 9138    .R cmr 9140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-omul 7025  df-er 7201  df-ec 7203  df-qs 7207  df-ni 9142  df-pli 9143  df-mi 9144  df-lti 9145  df-plpq 9178  df-mpq 9179  df-ltpq 9180  df-enq 9181  df-nq 9182  df-erq 9183  df-plq 9184  df-mq 9185  df-1nq 9186  df-rq 9187  df-ltnq 9188  df-np 9251  df-1p 9252  df-plp 9253  df-mp 9254  df-ltp 9255  df-mpr 9326  df-enr 9327  df-nr 9328  df-mr 9330  df-1r 9333  df-m1r 9334
This theorem is referenced by:  sqgt0sr  9374
  Copyright terms: Public domain W3C validator