Theorem m1p1sr 8092

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 8065	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2		df-1r 8064	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3	1, 2	oveq12i 6071	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4		df-0r 8063	. . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R
5		1pr 7886	. . . . 5 ⊢ 1P ∈ P
6		addclpr 7869	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
8		addsrpr 8077	. . . . 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 7911	. . . . . . 7 ⊢ ((1P ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)))
11	5, 5, 5, 10	mp3an 1374	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
12	11	oveq2i 6070	. . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
13		addclpr 7869	. . . . . . 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 7869	. . . . . . 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 8068	. . . . . 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 2258	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
21	4, 20	eqtr4i 2258	. 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
22	3, 21	eqtr4i 2258	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ⟨cop 3698  (class class class)co 6059  [cec 6779  Pcnp 7623  1_Pc1p 7624   +_P cpp 7625   ~_R cer 7628  0_Rc0r 7630  1_Rc1r 7631  -1_Rcm1r 7632   +_R cplr 7633

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-eprel 4416  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6781  df-ec 6783  df-qs 6787  df-ni 7636  df-pli 7637  df-mi 7638  df-lti 7639  df-plpq 7676  df-mpq 7677  df-enq 7679  df-nqqs 7680  df-plqqs 7681  df-mqqs 7682  df-1nqqs 7683  df-rq 7684  df-ltnqqs 7685  df-enq0 7756  df-nq0 7757  df-0nq0 7758  df-plq0 7759  df-mq0 7760  df-inp 7798  df-i1p 7799  df-iplp 7800  df-enr 8058  df-nr 8059  df-plr 8060  df-0r 8063  df-1r 8064  df-m1r 8065

This theorem is referenced by:  pn0sr  8103  ltm1sr  8109  caucvgsrlemoffres  8132  caucvgsr  8134  suplocsrlempr  8139  axi2m1  8207