Theorem m1p1sr 11007

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 10977	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2		df-1r 10976	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3	1, 2	oveq12i 7372	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
4		df-0r 10975	. . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R
5		1pr 10930	. . . . 5 ⊢ 1P ∈ P
6		addclpr 10933	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 693	. . . . 5 ⊢ (1P +P 1P) ∈ P
8		addsrpr 10990	. . . . 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 694	. . . 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 10937	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
11	10	oveq2i 7371	. . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
12		addclpr 10933	. . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
13	5, 7, 12	mp2an 693	. . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P
14		addclpr 10933	. . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P)
15	7, 5, 14	mp2an 693	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P
16		enreceq 10981	. . . . . 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 694	. . . . 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 2763	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
20	4, 19	eqtr4i 2763	. 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
21	3, 20	eqtr4i 2763	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ⟨cop 4587  (class class class)co 7360  [cec 8635  Pcnp 10774  1_Pc1p 10775   +_P cpp 10776   ~_R cer 10779  0_Rc0r 10781  1_Rc1r 10782  -1_Rcm1r 10783   +_R cplr 10784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-omul 8404  df-er 8637  df-ec 8639  df-qs 8643  df-ni 10787  df-pli 10788  df-mi 10789  df-lti 10790  df-plpq 10823  df-mpq 10824  df-ltpq 10825  df-enq 10826  df-nq 10827  df-erq 10828  df-plq 10829  df-mq 10830  df-1nq 10831  df-rq 10832  df-ltnq 10833  df-np 10896  df-1p 10897  df-plp 10898  df-ltp 10900  df-enr 10970  df-nr 10971  df-plr 10972  df-0r 10975  df-1r 10976  df-m1r 10977

This theorem is referenced by:  pn0sr  11016  supsrlem  11026  axi2m1  11074