Theorem m1m1sr 7828

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 7800	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2	1, 1	oveq12i 5934	. 2 ⊢ (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3		df-1r 7799	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
4		1pr 7621	. . . . 5 ⊢ 1P ∈ P
5		addclpr 7604	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
6	4, 4, 5	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
7		mulsrpr 7813	. . . . 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 7639	. . . . . . . . 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 7639	. . . . . . . . 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 7604	. . . . . . . 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 7646	. . . . . . 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 1348	. . . . . 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 7659	. . . . . . . . 9 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
18	4, 17	ax-mp 5	. . . . . . . 8 ⊢ (1P ·P 1P) = 1P
19		distrprg 7655	. . . . . . . . . 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 1348	. . . . . . . . 9 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
21		mulcomprg 7647	. . . . . . . . . . 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 5932	. . . . . . . . 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 2220	. . . . . . . 8 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
25	18, 24	oveq12i 5934	. . . . . . 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 5933	. . . . . 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 2220	. . . . 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 7639	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P ·P 1P) ∈ P)
29	4, 4, 28	mp2an 426	. . . . . . 7 ⊢ (1P ·P 1P) ∈ P
30		mulclpr 7639	. . . . . . . 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 7604	. . . . . . 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 7803	. . . . . 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 2220	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
38	3, 37	eqtr4i 2220	. 2 ⊢ 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
39	2, 38	eqtr4i 2220	1 ⊢ (-1R ·R -1R) = 1R

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ⟨cop 3625  (class class class)co 5922  [cec 6590  Pcnp 7358  1_Pc1p 7359   +_P cpp 7360   ·_P cmp 7361   ~_R cer 7363  1_Rc1r 7366  -1_Rcm1r 7367   ·_R cmr 7369

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-enr 7793  df-nr 7794  df-mr 7796  df-1r 7799  df-m1r 7800

This theorem is referenced by: (None)