Theorem m1m1sr 7969

Description: Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)

Assertion

Ref	Expression
m1m1sr	⊢ (-1R ·R -1R) = 1R

Proof of Theorem m1m1sr

Step	Hyp	Ref	Expression
1		df-m1r 7941	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2	1, 1	oveq12i 6023	. 2 ⊢ (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3		df-1r 7940	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
4		1pr 7762	. . . . 5 ⊢ 1P ∈ P
5		addclpr 7745	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
6	4, 4, 5	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
7		mulsrpr 7954	. . . . 5 ⊢ (((1P ∈ P ∧ (1P +P 1P) ∈ P) ∧ (1P ∈ P ∧ (1P +P 1P) ∈ P)) → ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R )
8	4, 6, 4, 6, 7	mp4an 427	. . . 4 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R
9		mulclpr 7780	. . . . . . . . 9 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
10	4, 6, 9	mp2an 426	. . . . . . . 8 ⊢ (1P ·P (1P +P 1P)) ∈ P
11		mulclpr 7780	. . . . . . . . 9 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) ·P 1P) ∈ P)
12	6, 4, 11	mp2an 426	. . . . . . . 8 ⊢ ((1P +P 1P) ·P 1P) ∈ P
13		addclpr 7745	. . . . . . . 8 ⊢ (((1P ·P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) ·P 1P) ∈ P) → ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P)
14	10, 12, 13	mp2an 426	. . . . . . 7 ⊢ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
15		addassprg 7787	. . . . . . 7 ⊢ ((1P ∈ P ∧ 1P ∈ P ∧ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P) → ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)))))
16	4, 4, 14, 15	mp3an 1371	. . . . . 6 ⊢ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))))
17		1idpr 7800	. . . . . . . . 9 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
18	4, 17	ax-mp 5	. . . . . . . 8 ⊢ (1P ·P 1P) = 1P
19		distrprg 7796	. . . . . . . . . 10 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P)))
20	6, 4, 4, 19	mp3an 1371	. . . . . . . . 9 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
21		mulcomprg 7788	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P))
22	4, 6, 21	mp2an 426	. . . . . . . . . 10 ⊢ (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
23	22	oveq1i 6021	. . . . . . . . 9 ⊢ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
24	20, 23	eqtr4i 2253	. . . . . . . 8 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
25	18, 24	oveq12i 6023	. . . . . . 7 ⊢ ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) = (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)))
26	25	oveq2i 6022	. . . . . 6 ⊢ (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P)))) = (1P +P (1P +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))))
27	16, 26	eqtr4i 2253	. . . . 5 ⊢ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))))
28		mulclpr 7780	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P ·P 1P) ∈ P)
29	4, 4, 28	mp2an 426	. . . . . . 7 ⊢ (1P ·P 1P) ∈ P
30		mulclpr 7780	. . . . . . . 8 ⊢ (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
31	6, 6, 30	mp2an 426	. . . . . . 7 ⊢ ((1P +P 1P) ·P (1P +P 1P)) ∈ P
32		addclpr 7745	. . . . . . 7 ⊢ (((1P ·P 1P) ∈ P ∧ ((1P +P 1P) ·P (1P +P 1P)) ∈ P) → ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P)
33	29, 31, 32	mp2an 426	. . . . . 6 ⊢ ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
34		enreceq 7944	. . . . . 6 ⊢ ((((1P +P 1P) ∈ P ∧ 1P ∈ P) ∧ (((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P ∧ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P)) → ([⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R ↔ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))))))
35	6, 4, 33, 14, 34	mp4an 427	. . . . 5 ⊢ ([⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R ↔ ((1P +P 1P) +P ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))) = (1P +P ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P)))))
36	27, 35	mpbir 146	. . . 4 ⊢ [⟨(1P +P 1P), 1P⟩] ~R = [⟨((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))), ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))⟩] ~R
37	8, 36	eqtr4i 2253	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
38	3, 37	eqtr4i 2253	. 2 ⊢ 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
39	2, 38	eqtr4i 2253	1 ⊢ (-1R ·R -1R) = 1R

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ⟨cop 3670  (class class class)co 6011  [cec 6693  Pcnp 7499  1_Pc1p 7500   +_P cpp 7501   ·_P cmp 7502   ~_R cer 7504  1_Rc1r 7507  -1_Rcm1r 7508   ·_R cmr 7510

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-eprel 4382  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-1o 6575  df-2o 6576  df-oadd 6579  df-omul 6580  df-er 6695  df-ec 6697  df-qs 6701  df-ni 7512  df-pli 7513  df-mi 7514  df-lti 7515  df-plpq 7552  df-mpq 7553  df-enq 7555  df-nqqs 7556  df-plqqs 7557  df-mqqs 7558  df-1nqqs 7559  df-rq 7560  df-ltnqqs 7561  df-enq0 7632  df-nq0 7633  df-0nq0 7634  df-plq0 7635  df-mq0 7636  df-inp 7674  df-i1p 7675  df-iplp 7676  df-imp 7677  df-enr 7934  df-nr 7935  df-mr 7937  df-1r 7940  df-m1r 7941

This theorem is referenced by: (None)