Theorem m1m1sr 5214

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 5129	. . . . 5 |- 1P e. P.
2		addclpr 5132	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
3	1, 1, 2	mp2an 699	. . . . 5 |- (1P +P. 1P) e. P.
4	1, 3	pm3.2i 285	. . . 4 |- (1P e. P. /\ (1P +P. 1P) e. P.)
5		mulsrpr 5197	. . . 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 699	. . 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 1821	. . . . . 6 |- 1P e. V
8		oprex 3989	. . . . . 6 |- ((1P .P. (1P +P. 1P)) +P. ((1P +P. 1P) .P. 1P)) e. V
9	7, 8	addasspr 5136	. . . . 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 5145	. . . . . . . 8 |- (1P e. P. -> (1P .P. 1P) = 1P)
11	1, 10	ax-mp 7	. . . . . . 7 |- (1P .P. 1P) = 1P
12	7, 7	distrpr 5144	. . . . . . . 8 |- ((1P +P. 1P) .P. (1P +P. 1P)) = (((1P +P. 1P) .P. 1P) +P. ((1P +P. 1P) .P. 1P))
13		oprex 3989	. . . . . . . . . 10 |- (1P +P. 1P) e. V
14	7, 13	mulcompr 5137	. . . . . . . . 9 |- (1P .P. (1P +P. 1P)) = ((1P +P. 1P) .P. 1P)
15	14	opreq1i 3977	. . . . . . . 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 1501	. . . . . . 7 |- ((1P +P. 1P) .P. (1P +P. 1P)) = ((1P .P. (1P +P. 1P)) +P. ((1P +P. 1P) .P. 1P))
17	11, 16	opreq12i 3979	. . . . . 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 3978	. . . . 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 1501	. . . 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 5134	. . . . . . . 8 |- ((1P e. P. /\ 1P e. P.) -> (1P .P. 1P) e. P.)
22	1, 1, 21	mp2an 699	. . . . . . 7 |- (1P .P. 1P) e. P.
23		mulclpr 5134	. . . . . . . 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 699	. . . . . . 7 |- ((1P +P. 1P) .P. (1P +P. 1P)) e. P.
25		addclpr 5132	. . . . . . 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 699	. . . . . 6 |- ((1P .P. 1P) +P. ((1P +P. 1P) .P. (1P +P. 1P))) e. P.
27		mulclpr 5134	. . . . . . . 8 |- ((1P e. P. /\ (1P +P. 1P) e. P.) -> (1P .P. (1P +P. 1P)) e. P.)
28	1, 3, 27	mp2an 699	. . . . . . 7 |- (1P .P. (1P +P. 1P)) e. P.
29		mulclpr 5134	. . . . . . . 8 |- (((1P +P. 1P) e. P. /\ 1P e. P.) -> ((1P +P. 1P) .P. 1P) e. P.)
30	3, 1, 29	mp2an 699	. . . . . . 7 |- ((1P +P. 1P) .P. 1P) e. P.
31		addclpr 5132	. . . . . . 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 699	. . . . . 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 5189	. . . . 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 699	. . . 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 1501	. 2 |- ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R ) = [<.(1P +P. 1P), 1P>.] ~R
38		df-m1r 5185	. . 3 |- -1R = [<.1P, (1P +P. 1P)>.] ~R
39	38, 38	opreq12i 3979	. 2 |- (-1R .R -1R) = ([<.1P, (1P +P. 1P)>.] ~R .R [<.1P, (1P +P. 1P)>.] ~R )
40		df-1r 5184	. 2 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
41	37, 39, 40	3eqtr4 1508	1 |- (-1R .R -1R) = 1R

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  <.cop 2415  (class class class)co 3969  [cec 4265  P.cnp 4997  1Pc1p 4998   +P. cpp 4999   .P. cmp 5000   ~R cer 5004  1Rc1r 5007  -1Rcm1r 5008   .R cmr 5010

This theorem is referenced by:  sqgt0sr 5227

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-mpr 5177  df-enr 5178  df-nr 5179  df-mr 5181  df-1r 5184  df-m1r 5185

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