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Theorem m1p1sr 7575
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
Assertion
Ref Expression
m1p1sr (-1R +R 1R) = 0R

Proof of Theorem m1p1sr
StepHypRef Expression
1 df-m1r 7548 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2 df-1r 7547 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
31, 2oveq12i 5786 . 2 (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4 df-0r 7546 . . 3 0R = [⟨1P, 1P⟩] ~R
5 1pr 7369 . . . . 5 1PP
6 addclpr 7352 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
75, 5, 6mp2an 422 . . . . 5 (1P +P 1P) ∈ P
8 addsrpr 7560 . . . . 5 (((1PP ∧ (1P +P 1P) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R )
95, 7, 7, 5, 8mp4an 423 . . . 4 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
10 addassprg 7394 . . . . . . 7 ((1PP ∧ 1PP ∧ 1PP) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)))
115, 5, 5, 10mp3an 1315 . . . . . 6 ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
1211oveq2i 5785 . . . . 5 (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
13 addclpr 7352 . . . . . . 7 ((1PP ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
145, 7, 13mp2an 422 . . . . . 6 (1P +P (1P +P 1P)) ∈ P
15 addclpr 7352 . . . . . . 7 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) +P 1P) ∈ P)
167, 5, 15mp2an 422 . . . . . 6 ((1P +P 1P) +P 1P) ∈ P
17 enreceq 7551 . . . . . 6 (((1PP ∧ 1PP) ∧ ((1P +P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) +P 1P) ∈ 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)))))
185, 5, 14, 16, 17mp4an 423 . . . . 5 ([⟨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))))
1912, 18mpbir 145 . . . 4 [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
209, 19eqtr4i 2163 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
214, 20eqtr4i 2163 . 2 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
223, 21eqtr4i 2163 1 (-1R +R 1R) = 0R
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  wcel 1480  cop 3530  (class class class)co 5774  [cec 6427  Pcnp 7106  1Pc1p 7107   +P cpp 7108   ~R cer 7111  0Rc0r 7113  1Rc1r 7114  -1Rcm1r 7115   +R cplr 7116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-i1p 7282  df-iplp 7283  df-enr 7541  df-nr 7542  df-plr 7543  df-0r 7546  df-1r 7547  df-m1r 7548
This theorem is referenced by:  pn0sr  7586  ltm1sr  7592  caucvgsrlemoffres  7615  caucvgsr  7617  suplocsrlempr  7622  axi2m1  7690
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