Theorem m1m1sr 5174

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 5089	. . . . 5 |- 1P e. P.
2		addclpr 5092	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
3	1, 1, 2	mp2an 695	. . . . 5 |- (1P +P. 1P) e. P.
4	1, 3	pm3.2i 285	. . . 4 |- (1P e. P. /\ (1P +P. 1P) e. P.)
5		mulsrpr 5157	. . . 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 695	. . 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 1809	. . . . . 6 |- 1P e. V
8		oprex 3968	. . . . . 6 |- ((1P .P. (1P +P. 1P)) +P. ((1P +P. 1P) .P. 1P)) e. V
9	7, 8	addasspr 5096	. . . . 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 5105	. . . . . . . 8 |- (1P e. P. -> (1P .P. 1P) = 1P)
11	1, 10	ax-mp 7	. . . . . . 7 |- (1P .P. 1P) = 1P
12	7, 7	distrpr 5104	. . . . . . . 8 |- ((1P +P. 1P) .P. (1P +P. 1P)) = (((1P +P. 1P) .P. 1P) +P. ((1P +P. 1P) .P. 1P))
13		oprex 3968	. . . . . . . . . 10 |- (1P +P. 1P) e. V
14	7, 13	mulcompr 5097	. . . . . . . . 9 |- (1P .P. (1P +P. 1P)) = ((1P +P. 1P) .P. 1P)
15	14	opreq1i 3956	. . . . . . . 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 1490	. . . . . . 7 |- ((1P +P. 1P) .P. (1P +P. 1P)) = ((1P .P. (1P +P. 1P)) +P. ((1P +P. 1P) .P. 1P))
17	11, 16	opreq12i 3958	. . . . . 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 3957	. . . . 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 1490	. . . 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 285	. . . . 5 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
21		mulclpr 5094	. . . . . . . 8 |- ((1P e. P. /\ 1P e. P.) -> (1P .P. 1P) e. P.)
22	1, 1, 21	mp2an 695	. . . . . . 7 |- (1P .P. 1P) e. P.
23		mulclpr 5094	. . . . . . . 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 695	. . . . . . 7 |- ((1P +P. 1P) .P. (1P +P. 1P)) e. P.
25		addclpr 5092	. . . . . . 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 695	. . . . . 6 |- ((1P .P. 1P) +P. ((1P +P. 1P) .P. (1P +P. 1P))) e. P.
27		mulclpr 5094	. . . . . . . 8 |- ((1P e. P. /\ (1P +P. 1P) e. P.) -> (1P .P. (1P +P. 1P)) e. P.)
28	1, 3, 27	mp2an 695	. . . . . . 7 |- (1P .P. (1P +P. 1P)) e. P.
29		mulclpr 5094	. . . . . . . 8 |- (((1P +P. 1P) e. P. /\ 1P e. P.) -> ((1P +P. 1P) .P. 1P) e. P.)
30	3, 1, 29	mp2an 695	. . . . . . 7 |- ((1P +P. 1P) .P. 1P) e. P.
31		addclpr 5092	. . . . . . 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 695	. . . . . 6 |- ((1P .P. (1P +P. 1P)) +P. ((1P +P. 1P) .P. 1P)) e. P.
33	26, 32	pm3.2i 285	. . . . 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 5149	. . . . 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 695	. . . 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 190	. . 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 1490	. 2 |- ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R ) = [<.(1P +P. 1P), 1P>.] ~R
38		df-m1r 5145	. . 3 |- -1R = [<.1P, (1P +P. 1P)>.] ~R
39	38, 38	opreq12i 3958	. 2 |- (-1R .R -1R) = ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R )
40		df-1r 5144	. 2 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
41	37, 39, 40	3eqtr4 1497	1 |- (-1R .R -1R) = 1R

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  <.cop 2401  (class class class)co 3948  [cec 4243  P.cnp 4957  1Pc1p 4958   +P. cpp 4959   .P. cmp 4960   ~R cer 4964  1Rc1r 4967  -1Rcm1r 4968   .R cmr 4970

This theorem is referenced by:  sqgt0sr 5187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-mpr 5137  df-enr 5138  df-nr 5139  df-mr 5141  df-1r 5144  df-m1r 5145

Copyright terms: Public domain