Theorem m1p1sr 7980

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

Step	Hyp	Ref	Expression
1		df-m1r 7953	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2		df-1r 7952	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3	1, 2	oveq12i 6030	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4		df-0r 7951	. . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R
5		1pr 7774	. . . . 5 ⊢ 1P ∈ P
6		addclpr 7757	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
8		addsrpr 7965	. . . . 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 427	. . . 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 7799	. . . . . . 7 ⊢ ((1P ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)))
11	5, 5, 5, 10	mp3an 1373	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
12	11	oveq2i 6029	. . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
13		addclpr 7757	. . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
14	5, 7, 13	mp2an 426	. . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P
15		addclpr 7757	. . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P)
16	7, 5, 15	mp2an 426	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P
17		enreceq 7956	. . . . . 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)))))
18	5, 5, 14, 16, 17	mp4an 427	. . . . 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))))
19	12, 18	mpbir 146	. . . 4 ⊢ [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
20	9, 19	eqtr4i 2255	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
21	4, 20	eqtr4i 2255	. 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
22	3, 21	eqtr4i 2255	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ⟨cop 3672  (class class class)co 6018  [cec 6700  Pcnp 7511  1_Pc1p 7512   +_P cpp 7513   ~_R cer 7516  0_Rc0r 7518  1_Rc1r 7519  -1_Rcm1r 7520   +_R cplr 7521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-enr 7946  df-nr 7947  df-plr 7948  df-0r 7951  df-1r 7952  df-m1r 7953

This theorem is referenced by:  pn0sr  7991  ltm1sr  7997  caucvgsrlemoffres  8020  caucvgsr  8022  suplocsrlempr  8027  axi2m1  8095