Theorem m1p1sr 11132

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

Step	Hyp	Ref	Expression
1		df-m1r 11102	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2		df-1r 11101	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3	1, 2	oveq12i 7443	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4		df-0r 11100	. . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R
5		1pr 11055	. . . . 5 ⊢ 1P ∈ P
6		addclpr 11058	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 692	. . . . 5 ⊢ (1P +P 1P) ∈ P
8		addsrpr 11115	. . . . 5 ⊢ (((1P ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R )
9	5, 7, 7, 5, 8	mp4an 693	. . . 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 11062	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
11	10	oveq2i 7442	. . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
12		addclpr 11058	. . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
13	5, 7, 12	mp2an 692	. . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P
14		addclpr 11058	. . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P)
15	7, 5, 14	mp2an 692	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P
16		enreceq 11106	. . . . . 6 ⊢ (((1P ∈ P ∧ 1P ∈ P) ∧ ((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)))))
17	5, 5, 13, 15, 16	mp4an 693	. . . . 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))))
18	11, 17	mpbir 231	. . . 4 ⊢ [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
19	9, 18	eqtr4i 2768	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
20	4, 19	eqtr4i 2768	. 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
21	3, 20	eqtr4i 2768	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ⟨cop 4632  (class class class)co 7431  [cec 8743  Pcnp 10899  1_Pc1p 10900   +_P cpp 10901   ~_R cer 10904  0_Rc0r 10906  1_Rc1r 10907  -1_Rcm1r 10908   +_R cplr 10909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958  df-np 11021  df-1p 11022  df-plp 11023  df-ltp 11025  df-enr 11095  df-nr 11096  df-plr 11097  df-0r 11100  df-1r 11101  df-m1r 11102

This theorem is referenced by:  pn0sr  11141  supsrlem  11151  axi2m1  11199