Theorem m1p1sr 10847

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 10817	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2		df-1r 10816	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3	1, 2	oveq12i 7281	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4		df-0r 10815	. . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R
5		1pr 10770	. . . . 5 ⊢ 1P ∈ P
6		addclpr 10773	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 689	. . . . 5 ⊢ (1P +P 1P) ∈ P
8		addsrpr 10830	. . . . 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 690	. . . 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 10777	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
11	10	oveq2i 7280	. . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
12		addclpr 10773	. . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
13	5, 7, 12	mp2an 689	. . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P
14		addclpr 10773	. . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P)
15	7, 5, 14	mp2an 689	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P
16		enreceq 10821	. . . . . 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 690	. . . . 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 230	. . . 4 ⊢ [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
19	9, 18	eqtr4i 2771	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
20	4, 19	eqtr4i 2771	. 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
21	3, 20	eqtr4i 2771	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2110  ⟨cop 4573  (class class class)co 7269  [cec 8477  Pcnp 10614  1_Pc1p 10615   +_P cpp 10616   ~_R cer 10619  0_Rc0r 10621  1_Rc1r 10622  -1_Rcm1r 10623   +_R cplr 10624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-inf2 9375

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-oadd 8290  df-omul 8291  df-er 8479  df-ec 8481  df-qs 8485  df-ni 10627  df-pli 10628  df-mi 10629  df-lti 10630  df-plpq 10663  df-mpq 10664  df-ltpq 10665  df-enq 10666  df-nq 10667  df-erq 10668  df-plq 10669  df-mq 10670  df-1nq 10671  df-rq 10672  df-ltnq 10673  df-np 10736  df-1p 10737  df-plp 10738  df-ltp 10740  df-enr 10810  df-nr 10811  df-plr 10812  df-0r 10815  df-1r 10816  df-m1r 10817

This theorem is referenced by:  pn0sr  10856  supsrlem  10866  axi2m1  10914