Theorem m1m1sr 6846

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 6818	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2	1, 1	oveq12i 5524	. 2 ⊢ (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3		df-1r 6817	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
4		1pr 6652	. . . . 5 ⊢ 1P ∈ P
5		addclpr 6635	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
6	4, 4, 5	mp2an 402	. . . . 5 ⊢ (1P +P 1P) ∈ P
7		mulsrpr 6831	. . . . 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 403	. . . 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 6670	. . . . . . . . 9 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
10	4, 6, 9	mp2an 402	. . . . . . . 8 ⊢ (1P ·P (1P +P 1P)) ∈ P
11		mulclpr 6670	. . . . . . . . 9 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) ·P 1P) ∈ P)
12	6, 4, 11	mp2an 402	. . . . . . . 8 ⊢ ((1P +P 1P) ·P 1P) ∈ P
13		addclpr 6635	. . . . . . . 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 402	. . . . . . 7 ⊢ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
15		addassprg 6677	. . . . . . 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 1232	. . . . . 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 6690	. . . . . . . . 9 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
18	4, 17	ax-mp 7	. . . . . . . 8 ⊢ (1P ·P 1P) = 1P
19		distrprg 6686	. . . . . . . . . 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 1232	. . . . . . . . 9 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
21		mulcomprg 6678	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P))
22	4, 6, 21	mp2an 402	. . . . . . . . . 10 ⊢ (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
23	22	oveq1i 5522	. . . . . . . . 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 2063	. . . . . . . 8 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
25	18, 24	oveq12i 5524	. . . . . . 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 5523	. . . . . 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 2063	. . . . 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 6670	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P ·P 1P) ∈ P)
29	4, 4, 28	mp2an 402	. . . . . . 7 ⊢ (1P ·P 1P) ∈ P
30		mulclpr 6670	. . . . . . . 8 ⊢ (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
31	6, 6, 30	mp2an 402	. . . . . . 7 ⊢ ((1P +P 1P) ·P (1P +P 1P)) ∈ P
32		addclpr 6635	. . . . . . 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 402	. . . . . 6 ⊢ ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
34		enreceq 6821	. . . . . 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 403	. . . . 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 134	. . . 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 2063	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
38	3, 37	eqtr4i 2063	. 2 ⊢ 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
39	2, 38	eqtr4i 2063	1 ⊢ (-1R ·R -1R) = 1R

Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  (class class class)co 5512  [cec 6104  Pcnp 6389  1_Pc1p 6390   +_P cpp 6391   ·_P cmp 6392   ~_R cer 6394  1_Rc1r 6397  -1_Rcm1r 6398   ·_R cmr 6400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-enr 6811  df-nr 6812  df-mr 6814  df-1r 6817  df-m1r 6818

This theorem is referenced by: (None)