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Theorem m1m1sr 9500
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 9470 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21, 1oveq12i 6290 . 2  |-  ( -1R 
.R  -1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
3 df-1r 9469 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
4 1pr 9423 . . . . 5  |-  1P  e.  P.
5 addclpr 9426 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
64, 4, 5mp2an 670 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
7 mulsrpr 9483 . . . . 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 671 . . . 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 9430 . . . . . 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 9437 . . . . . . . . 9  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
114, 10ax-mp 5 . . . . . . . 8  |-  ( 1P 
.P.  1P )  =  1P
12 distrpr 9436 . . . . . . . . 9  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( ( 1P  +P.  1P )  .P.  1P )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
13 mulcompr 9431 . . . . . . . . . 10  |-  ( 1P 
.P.  ( 1P  +P.  1P ) )  =  ( ( 1P  +P.  1P )  .P.  1P )
1413oveq1i 6288 . . . . . . . . 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 2434 . . . . . . . 8  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( 1P  .P.  ( 1P  +P.  1P ) )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
1611, 15oveq12i 6290 . . . . . . 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 6289 . . . . . 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 2434 . . . . 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 9428 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  .P.  1P )  e.  P. )
204, 4, 19mp2an 670 . . . . . . 7  |-  ( 1P 
.P.  1P )  e.  P.
21 mulclpr 9428 . . . . . . . 8  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  ( 1P  +P.  1P )  e. 
P. )  ->  (
( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  e.  P. )
226, 6, 21mp2an 670 . . . . . . 7  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  e.  P.
23 addclpr 9426 . . . . . . 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 670 . . . . . 6  |-  ( ( 1P  .P.  1P )  +P.  ( ( 1P 
+P.  1P )  .P.  ( 1P  +P.  1P ) ) )  e.  P.
25 mulclpr 9428 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( 1P  .P.  ( 1P  +P.  1P ) )  e.  P. )
264, 6, 25mp2an 670 . . . . . . 7  |-  ( 1P 
.P.  ( 1P  +P.  1P ) )  e.  P.
27 mulclpr 9428 . . . . . . . 8  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  1P )  .P.  1P )  e. 
P. )
286, 4, 27mp2an 670 . . . . . . 7  |-  ( ( 1P  +P.  1P )  .P.  1P )  e. 
P.
29 addclpr 9426 . . . . . . 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 670 . . . . . 6  |-  ( ( 1P  .P.  ( 1P 
+P.  1P ) )  +P.  ( ( 1P  +P.  1P )  .P.  1P ) )  e.  P.
31 enreceq 9473 . . . . . 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 671 . . . . 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 2434 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  .R 
[ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
353, 34eqtr4i 2434 . 2  |-  1R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
362, 35eqtr4i 2434 1  |-  ( -1R 
.R  -1R )  =  1R
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842   <.cop 3978  (class class class)co 6278   [cec 7346   P.cnp 9267   1Pc1p 9268    +P. cpp 9269    .P. cmp 9270    ~R cer 9272   1Rc1r 9275   -1Rcm1r 9276    .R cmr 9278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-omul 7172  df-er 7348  df-ec 7350  df-qs 7354  df-ni 9280  df-pli 9281  df-mi 9282  df-lti 9283  df-plpq 9316  df-mpq 9317  df-ltpq 9318  df-enq 9319  df-nq 9320  df-erq 9321  df-plq 9322  df-mq 9323  df-1nq 9324  df-rq 9325  df-ltnq 9326  df-np 9389  df-1p 9390  df-plp 9391  df-mp 9392  df-ltp 9393  df-enr 9463  df-nr 9464  df-mr 9466  df-1r 9469  df-m1r 9470
This theorem is referenced by:  sqgt0sr  9513
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