MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1p1sr Structured version   Visualization version   GIF version

Theorem m1p1sr 11121
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
m1p1sr (-1R +R 1R) = 0R

Proof of Theorem m1p1sr
StepHypRef Expression
1 df-m1r 11091 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2 df-1r 11090 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
31, 2oveq12i 7436 . 2 (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4 df-0r 11089 . . 3 0R = [⟨1P, 1P⟩] ~R
5 1pr 11044 . . . . 5 1PP
6 addclpr 11047 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
75, 5, 6mp2an 690 . . . . 5 (1P +P 1P) ∈ P
8 addsrpr 11104 . . . . 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 691 . . . 4 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
10 addasspr 11051 . . . . . 6 ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
1110oveq2i 7435 . . . . 5 (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
12 addclpr 11047 . . . . . . 7 ((1PP ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
135, 7, 12mp2an 690 . . . . . 6 (1P +P (1P +P 1P)) ∈ P
14 addclpr 11047 . . . . . . 7 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) +P 1P) ∈ P)
157, 5, 14mp2an 690 . . . . . 6 ((1P +P 1P) +P 1P) ∈ P
16 enreceq 11095 . . . . . 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)))))
175, 5, 13, 15, 16mp4an 691 . . . . 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))))
1811, 17mpbir 230 . . . 4 [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
199, 18eqtr4i 2758 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
204, 19eqtr4i 2758 . 2 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
213, 20eqtr4i 2758 1 (-1R +R 1R) = 0R
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  cop 4636  (class class class)co 7424  [cec 8727  Pcnp 10888  1Pc1p 10889   +P cpp 10890   ~R cer 10893  0Rc0r 10895  1Rc1r 10896  -1Rcm1r 10897   +R cplr 10898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-inf2 9670
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-oadd 8495  df-omul 8496  df-er 8729  df-ec 8731  df-qs 8735  df-ni 10901  df-pli 10902  df-mi 10903  df-lti 10904  df-plpq 10937  df-mpq 10938  df-ltpq 10939  df-enq 10940  df-nq 10941  df-erq 10942  df-plq 10943  df-mq 10944  df-1nq 10945  df-rq 10946  df-ltnq 10947  df-np 11010  df-1p 11011  df-plp 11012  df-ltp 11014  df-enr 11084  df-nr 11085  df-plr 11086  df-0r 11089  df-1r 11090  df-m1r 11091
This theorem is referenced by:  pn0sr  11130  supsrlem  11140  axi2m1  11188
  Copyright terms: Public domain W3C validator