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Theorem m1p1sr 6353
Description: Minus one plus one is zero for signed reals.
Assertion
Ref Expression
m1p1sr |- (-1R +R 1R) = 0R
Proof of Theorem m1p1sr
StepHypRef Expression
1 1pr 6269 . . . . 5 |- 1P e. P.
2 addclpr 6272 . . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
31, 1, 2mp2an 761 . . . . 5 |- (1P +P. 1P) e. P.
41, 3pm3.2i 307 . . . 4 |- (1P e. P. /\ (1P +P. 1P) e. P.)
53, 1pm3.2i 307 . . . 4 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
6 addsrpr 6336 . . . 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 )
74, 5, 6mp2an 761 . . 3 |- ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R ) = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R
81elisseti 2301 . . . . . 6 |- 1P e. _V
98, 8addasspr 6276 . . . . 5 |- ((1P +P. 1P) +P. 1P) = (1P +P. (1P +P. 1P))
109opreq2i 4893 . . . 4 |- (1P +P. ((1P +P. 1P) +P. 1P)) = (1P +P. (1P +P. (1P +P. 1P)))
111, 1pm3.2i 307 . . . . 5 |- (1P e. P. /\ 1P e. P.)
12 addclpr 6272 . . . . . . 7 |- ((1P e. P. /\ (1P +P. 1P) e. P.) -> (1P +P. (1P +P. 1P)) e. P.)
131, 3, 12mp2an 761 . . . . . 6 |- (1P +P. (1P +P. 1P)) e. P.
14 addclpr 6272 . . . . . . 7 |- (((1P +P. 1P) e. P. /\ 1P e. P.) -> ((1P +P. 1P) +P. 1P) e. P.)
153, 1, 14mp2an 761 . . . . . 6 |- ((1P +P. 1P) +P. 1P) e. P.
1613, 15pm3.2i 307 . . . . 5 |- ((1P +P. (1P +P. 1P)) e. P. /\ ((1P +P. 1P) +P. 1P) e. P.)
17 enreceq 6329 . . . . 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)))))
1811, 16, 17mp2an 761 . . . 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))))
1910, 18mpbir 207 . . 3 |- [<.1P, 1P>.] ~R = [<.(1P +P. (1P +P. 1P)), ((1P +P. 1P) +P. 1P)>.] ~R
207, 19eqtr4i 1911 . 2 |- ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R ) = [<.1P, 1P>.] ~R
21 df-m1r 6325 . . 3 |- -1R = [<.1P, (1P +P. 1P)>.] ~R
22 df-1r 6324 . . 3 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
2321, 22opreq12i 4894 . 2 |- (-1R +R 1R) = ([<.1P, (1P +P. 1P)>.] ~R +R [<.(1P +P. 1P), 1P>.] ~R )
24 df-0r 6323 . 2 |- 0R = [<.1P, 1P>.] ~R
2520, 23, 243eqtr4i 1921 1 |- (-1R +R 1R) = 0R
Colors of variables: wff set class
Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  <.cop 3046  (class class class)co 4884  [cec 5316  P.cnp 6137  1Pc1p 6138   +P. cpp 6139   ~R cer 6144  0Rc0r 6146  1Rc1r 6147  -1Rcm1r 6148   +R cplr 6149
This theorem is referenced by:  pn0sr 6362  supsrlem5 6381  axi2m1 6438
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-ltp 6242  df-plpr 6316  df-enr 6318  df-nr 6319  df-plr 6320  df-0r 6323  df-1r 6324  df-m1r 6325
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