Theorem m1p1sr 3995

Description: Minus one plus one is zero for signed reals.

Assertion

Ref	Expression
m1p1sr	⊢ (-1R +R 1R) = 0R

Proof of Theorem m1p1sr

Step	Hyp	Ref	Expression
1		1pr 3911	. . . . 5 ⊢ 1P ∈ P
2		addclpr 3914	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
3	1, 1, 2	mp2an 520	. . . . 5 ⊢ (1P +P 1P) ∈ P
4	1, 3	pm3.2i 234	. . . 4 ⊢ (1P ∈ P ∧ (1P +P 1P) ∈ P)
5	3, 1	pm3.2i 234	. . . 4 ⊢ ((1P +P 1P) ∈ P ∧ 1P ∈ P)
6		addsrpr 3978	. . . 4 ⊢ (((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 )
7	4, 5, 6	mp2an 520	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
8	1	elisseti 1355	. . . . . 6 ⊢ 1P ∈ V
9	8, 8	addasspr 3918	. . . . 5 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))
10	9	opreq2i 3010	. . . 4 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))
11	1, 1	pm3.2i 234	. . . . 5 ⊢ (1P ∈ P ∧ 1P ∈ P)
12		addclpr 3914	. . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P)
13	1, 3, 12	mp2an 520	. . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P
14		addclpr 3914	. . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P)
15	3, 1, 14	mp2an 520	. . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P
16	13, 15	pm3.2i 234	. . . . 5 ⊢ ((1P +P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) +P 1P) ∈ P)
17		enreceq 3971	. . . . 5 ⊢ (((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	11, 16, 17	mp2an 520	. . . 4 ⊢ ([⟨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	10, 18	mpbir 165	. . 3 ⊢ [⟨1P, 1P⟩] ~R = [⟨(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)⟩] ~R
20	7, 19	eqtr4 1122	. 2 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R
21		df-m1r 3967	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
22		df-1r 3966	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
23	21, 22	opreq12i 3011	. 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
24		df-0r 3965	. 2 ⊢ 0R = [⟨1P, 1P⟩] ~R
25	20, 23, 24	3eqtr4 1126	1 ⊢ (-1R +R 1R) = 0R

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  [cec 3198  Pcnp 3779  1_Pc1p 3780   +_P cpp 3781   ~_R cer 3786  0_Rc0r 3788  1_Rc1r 3789  -1_Rcm1r 3790   +_R cplr 3791

This theorem is referenced by:  pn0sr 4004  supsrlem5 4023  axi2m1 4082

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-m1r 3967

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