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Theorem m1m1sr 9466
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 9436 . . 3  |-  -1R  =  [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R
21, 1oveq12i 6294 . 2  |-  ( -1R 
.R  -1R )  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
3 df-1r 9435 . . 3  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
4 1pr 9389 . . . . 5  |-  1P  e.  P.
5 addclpr 9392 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
64, 4, 5mp2an 672 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
7 mulsrpr 9449 . . . . 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 9396 . . . . . 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 9403 . . . . . . . . 9  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
114, 10ax-mp 5 . . . . . . . 8  |-  ( 1P 
.P.  1P )  =  1P
12 distrpr 9402 . . . . . . . . 9  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( ( 1P  +P.  1P )  .P.  1P )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
13 mulcompr 9397 . . . . . . . . . 10  |-  ( 1P 
.P.  ( 1P  +P.  1P ) )  =  ( ( 1P  +P.  1P )  .P.  1P )
1413oveq1i 6292 . . . . . . . . 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 2499 . . . . . . . 8  |-  ( ( 1P  +P.  1P )  .P.  ( 1P  +P.  1P ) )  =  ( ( 1P  .P.  ( 1P  +P.  1P ) )  +P.  ( ( 1P 
+P.  1P )  .P.  1P ) )
1611, 15oveq12i 6294 . . . . . . 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 6293 . . . . . 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 2499 . . . . 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 9394 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  .P.  1P )  e.  P. )
204, 4, 19mp2an 672 . . . . . . 7  |-  ( 1P 
.P.  1P )  e.  P.
21 mulclpr 9394 . . . . . . . 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 9392 . . . . . . 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 9394 . . . . . . . 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 9394 . . . . . . . 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 9392 . . . . . . 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 9439 . . . . . 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 2499 . . 3  |-  ( [
<. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  .R 
[ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
353, 34eqtr4i 2499 . 2  |-  1R  =  ( [ <. 1P ,  ( 1P  +P.  1P )
>. ]  ~R  .R  [ <. 1P ,  ( 1P 
+P.  1P ) >. ]  ~R  )
362, 35eqtr4i 2499 1  |-  ( -1R 
.R  -1R )  =  1R
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   <.cop 4033  (class class class)co 6282   [cec 7306   P.cnp 9233   1Pc1p 9234    +P. cpp 9235    .P. cmp 9236    ~R cer 9238   1Rc1r 9241   -1Rcm1r 9242    .R cmr 9244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-1p 9356  df-plp 9357  df-mp 9358  df-ltp 9359  df-enr 9429  df-nr 9430  df-mr 9432  df-1r 9435  df-m1r 9436
This theorem is referenced by:  sqgt0sr  9479
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