Theorem m1p1sr 5184

Description: Minus one plus one is zero for signed reals.

Assertion

Ref	Expression
m1p1sr	|- (-1R +R 1R) = 0R

Proof of Theorem m1p1sr

Step	Hyp	Ref	Expression
1		1pr 5100	. . . . 5 |- 1P e. P.
2		addclpr 5103	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
3	1, 1, 2	mp2an 696	. . . . 5 |- (1P +P. 1P) e. P.
4	1, 3	pm3.2i 285	. . . 4 |- (1P e. P. /\ (1P +P. 1P) e. P.)
5	3, 1	pm3.2i 285	. . . 4 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
6		addsrpr 5167	. . . 4 |- (((1P e. P. /\ (1P +P. 1P) e. P.) /\ ((1P +P. 1P) e. P. /\ 1P e. P.)) -> ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R ) = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R )
7	4, 5, 6	mp2an 696	. . 3 |- ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R ) = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R
8	1	elisseti 1815	. . . . . 6 |- 1P e. V
9	8, 8	addasspr 5107	. . . . 5 |- ((1P +P. 1P) +P. 1P) = (1P +P. (1P +P. 1P))
10	9	opreq2i 3967	. . . 4 |- (1P +P. ((1P +P. 1P) +P. 1P)) = (1P +P. (1P +P. (1P +P. 1P)))
11	1, 1	pm3.2i 285	. . . . 5 |- (1P e. P. /\ 1P e. P.)
12		addclpr 5103	. . . . . . 7 |- ((1P e. P. /\ (1P +P. 1P) e. P.) -> (1P +P. (1P +P. 1P)) e. P.)
13	1, 3, 12	mp2an 696	. . . . . 6 |- (1P +P. (1P +P. 1P)) e. P.
14		addclpr 5103	. . . . . . 7 |- (((1P +P. 1P) e. P. /\ 1P e. P.) -> ((1P +P. 1P) +P. 1P) e. P.)
15	3, 1, 14	mp2an 696	. . . . . 6 |- ((1P +P. 1P) +P. 1P) e. P.
16	13, 15	pm3.2i 285	. . . . 5 |- ((1P +P. (1P +P. 1P)) e. P. /\ ((1P +P. 1P) +P. 1P) e. P.)
17		enreceq 5160	. . . . 5 |- (((1P e. P. /\ 1P e. P.) /\ ((1P +P. (1P +P. 1P)) e. P. /\ ((1P +P. 1P) +P. 1P) e. P.)) -> ([<.1P, 1P>.] ~R = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R <-> (1P +P. ((1P +P. 1P) +P. 1P)) = (1P +P. (1P +P. (1P +P. 1P)))))
18	11, 16, 17	mp2an 696	. . . 4 |- ([<.1P, 1P>.] ~R = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R <-> (1P +P. ((1P +P. 1P) +P. 1P)) = (1P +P. (1P +P. (1P +P. 1P))))
19	10, 18	mpbir 190	. . 3 |- [<.1P, 1P>.] ~R = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R
20	7, 19	eqtr4 1496	. 2 |- ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R ) = [<.1P, 1P>.] ~R
21		df-m1r 5156	. . 3 |- -1R = [<.1P, (1P +P. 1P)>.] ~R
22		df-1r 5155	. . 3 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
23	21, 22	opreq12i 3968	. 2 |- (-1R +R 1R) = ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R )
24		df-0r 5154	. 2 |- 0R = [<.1P, 1P>.] ~R
25	20, 23, 24	3eqtr4 1503	1 |- (-1R +R 1R) = 0R

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  <.cop 2408  (class class class)co 3958  [cec 4252  P.cnp 4968  1Pc1p 4969   +P. cpp 4970   ~R cer 4975  0Rc0r 4977  1Rc1r 4978  -1Rcm1r 4979   +R cplr 4980

This theorem is referenced by:  pn0sr 5193  supsrlem5 5212  axi2m1 5268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-ltp 5073  df-plpr 5147  df-enr 5149  df-nr 5150  df-plr 5151  df-0r 5154  df-1r 5155  df-m1r 5156

Copyright terms: Public domain