Theorem m1m1sr 3996

Description: Minus one times minus one is plus one for signed reals.

Assertion

Ref	Expression
m1m1sr	|- (-1R .R -1R) = 1R

Proof of Theorem m1m1sr

Step	Hyp	Ref	Expression
1		1pr 3911	. . . . 5 |- 1P e. P.
2		addclpr 3914	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
3	1, 1, 2	mp2an 520	. . . . 5 |- (1P +P. 1P) e. P.
4	1, 3	pm3.2i 234	. . . 4 |- (1P e. P. /\ (1P +P. 1P) e. P.)
5		mulsrpr 3979	. . . 4 |- (((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 )
6	4, 4, 5	mp2an 520	. . 3 |- ([<.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
7	1	elisseti 1355	. . . . . 6 |- 1P e. V
8		oprex 3018	. . . . . 6 |- ((1P .P (1P +P. 1P)) +P. ((1P +P. 1P) .P 1P)) e. V
9	7, 8	addasspr 3918	. . . . 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) .P 1P))))
10		1idpr 3927	. . . . . . . 8 |- (1P e. P. -> (1P .P 1P) = 1P)
11	1, 10	ax-mp 6	. . . . . . 7 |- (1P .P 1P) = 1P
12	7, 7	distrpr 3926	. . . . . . . 8 |- ((1P +P. 1P) .P (1P +P. 1P)) = (((1P +P. 1P) .P 1P) +P. ((1P +P. 1P) .P 1P))
13		oprex 3018	. . . . . . . . . 10 |- (1P +P. 1P) e. V
14	7, 13	mulcompr 3919	. . . . . . . . 9 |- (1P .P (1P +P. 1P)) = ((1P +P. 1P) .P 1P)
15	14	opreq1i 3009	. . . . . . . 8 |- ((1P .P (1P +P. 1P)) +P. ((1P +P. 1P) .P 1P)) = (((1P +P. 1P) .P 1P) +P. ((1P +P. 1P) .P 1P))
16	12, 15	eqtr4 1122	. . . . . . 7 |- ((1P +P. 1P) .P (1P +P. 1P)) = ((1P .P (1P +P. 1P)) +P. ((1P +P. 1P) .P 1P))
17	11, 16	opreq12i 3011	. . . . . 6 |- ((1P .P 1P) +P. ((1P +P. 1P) .P (1P +P. 1P))) = (1P +P. ((1P .P (1P +P. 1P)) +P. ((1P +P. 1P) .P 1P)))
18	17	opreq2i 3010	. . . . 5 |- (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))))
19	9, 18	eqtr4 1122	. . . 4 |- ((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))))
20	3, 1	pm3.2i 234	. . . . 5 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
21		mulclpr 3916	. . . . . . . 8 |- ((1P e. P. /\ 1P e. P.) -> (1P .P 1P) e. P.)
22	1, 1, 21	mp2an 520	. . . . . . 7 |- (1P .P 1P) e. P.
23		mulclpr 3916	. . . . . . . 8 |- (((1P +P. 1P) e. P. /\ (1P +P. 1P) e. P.) -> ((1P +P. 1P) .P (1P +P. 1P)) e. P.)
24	3, 3, 23	mp2an 520	. . . . . . 7 |- ((1P +P. 1P) .P (1P +P. 1P)) e. P.
25		addclpr 3914	. . . . . . 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.)
26	22, 24, 25	mp2an 520	. . . . . 6 |- ((1P .P 1P) +P. ((1P +P. 1P) .P (1P +P. 1P))) e. P.
27		mulclpr 3916	. . . . . . . 8 |- ((1P e. P. /\ (1P +P. 1P) e. P.) -> (1P .P (1P +P. 1P)) e. P.)
28	1, 3, 27	mp2an 520	. . . . . . 7 |- (1P .P (1P +P. 1P)) e. P.
29		mulclpr 3916	. . . . . . . 8 |- (((1P +P. 1P) e. P. /\ 1P e. P.) -> ((1P +P. 1P) .P 1P) e. P.)
30	3, 1, 29	mp2an 520	. . . . . . 7 |- ((1P +P. 1P) .P 1P) e. P.
31		addclpr 3914	. . . . . . 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.)
32	28, 30, 31	mp2an 520	. . . . . 6 |- ((1P .P (1P +P. 1P)) +P. ((1P +P. 1P) .P 1P)) e. P.
33	26, 32	pm3.2i 234	. . . . 5 |- (((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.)
34		enreceq 3971	. . . . 5 |- ((((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))))))
35	20, 33, 34	mp2an 520	. . . 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 <-> ((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)))))
36	19, 35	mpbir 165	. . 3 |- [<.(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
37	6, 36	eqtr4 1122	. 2 |- ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R ) = [<.(1P +P. 1P), 1P>.] ~R
38		df-m1r 3967	. . 3 |- -1R = [<.1P, (1P +P. 1P)>.] ~R
39	38, 38	opreq12i 3011	. 2 |- (-1R .R -1R) = ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R )
40		df-1r 3966	. 2 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
41	37, 39, 40	3eqtr4 1126	1 |- (-1R .R -1R) = 1R

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810  (class class class)co 3001  [cec 3198  P.cnp 3779  1Pc1p 3780   +P. cpp 3781   .P cmp 3782   ~R cer 3786  1Rc1r 3789  -1Rcm1r 3790   .R cmr 3792

This theorem is referenced by:  sqgt0sr 4009  axrecex 4079

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-mpr 3959  df-enr 3960  df-nr 3961  df-mr 3963  df-1r 3966  df-m1r 3967

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