Theorem m1m1sr 7823

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 7795	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2	1, 1	oveq12i 5931	. 2 ⊢ (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3		df-1r 7794	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
4		1pr 7616	. . . . 5 ⊢ 1P ∈ P
5		addclpr 7599	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
6	4, 4, 5	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
7		mulsrpr 7808	. . . . 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 7634	. . . . . . . . 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 7634	. . . . . . . . 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 7599	. . . . . . . 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 7641	. . . . . . 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 7654	. . . . . . . . 9 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
18	4, 17	ax-mp 5	. . . . . . . 8 ⊢ (1P ·P 1P) = 1P
19		distrprg 7650	. . . . . . . . . 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 7642	. . . . . . . . . . 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 5929	. . . . . . . . 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 2217	. . . . . . . 8 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
25	18, 24	oveq12i 5931	. . . . . . 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 5930	. . . . . 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 2217	. . . . 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 7634	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P ·P 1P) ∈ P)
29	4, 4, 28	mp2an 426	. . . . . . 7 ⊢ (1P ·P 1P) ∈ P
30		mulclpr 7634	. . . . . . . 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 7599	. . . . . . 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 7798	. . . . . 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 2217	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
38	3, 37	eqtr4i 2217	. 2 ⊢ 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
39	2, 38	eqtr4i 2217	1 ⊢ (-1R ·R -1R) = 1R

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ⟨cop 3622  (class class class)co 5919  [cec 6587  Pcnp 7353  1_Pc1p 7354   +_P cpp 7355   ·_P cmp 7356   ~_R cer 7358  1_Rc1r 7361  -1_Rcm1r 7362   ·_R cmr 7364

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-imp 7531  df-enr 7788  df-nr 7789  df-mr 7791  df-1r 7794  df-m1r 7795

This theorem is referenced by: (None)