Theorem m1m1sr 10984

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

Assertion

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

Proof of Theorem m1m1sr

Step	Hyp	Ref	Expression
1		df-m1r 10953	. . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
2	1, 1	oveq12i 7358	. 2 ⊢ (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3		df-1r 10952	. . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
4		1pr 10906	. . . . 5 ⊢ 1P ∈ P
5		addclpr 10909	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
6	4, 4, 5	mp2an 692	. . . . 5 ⊢ (1P +P 1P) ∈ P
7		mulsrpr 10967	. . . . 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 693	. . . 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		addasspr 10913	. . . . . 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))))
10		1idpr 10920	. . . . . . . . 9 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
11	4, 10	ax-mp 5	. . . . . . . 8 ⊢ (1P ·P 1P) = 1P
12		distrpr 10919	. . . . . . . . 9 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
13		mulcompr 10914	. . . . . . . . . 10 ⊢ (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
14	13	oveq1i 7356	. . . . . . . . 9 ⊢ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
15	12, 14	eqtr4i 2757	. . . . . . . 8 ⊢ ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
16	11, 15	oveq12i 7358	. . . . . . 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)))
17	16	oveq2i 7357	. . . . . 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))))
18	9, 17	eqtr4i 2757	. . . . 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))))
19		mulclpr 10911	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P ·P 1P) ∈ P)
20	4, 4, 19	mp2an 692	. . . . . . 7 ⊢ (1P ·P 1P) ∈ P
21		mulclpr 10911	. . . . . . . 8 ⊢ (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
22	6, 6, 21	mp2an 692	. . . . . . 7 ⊢ ((1P +P 1P) ·P (1P +P 1P)) ∈ P
23		addclpr 10909	. . . . . . 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)
24	20, 22, 23	mp2an 692	. . . . . 6 ⊢ ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
25		mulclpr 10911	. . . . . . . 8 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
26	4, 6, 25	mp2an 692	. . . . . . 7 ⊢ (1P ·P (1P +P 1P)) ∈ P
27		mulclpr 10911	. . . . . . . 8 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) ·P 1P) ∈ P)
28	6, 4, 27	mp2an 692	. . . . . . 7 ⊢ ((1P +P 1P) ·P 1P) ∈ P
29		addclpr 10909	. . . . . . 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)
30	26, 28, 29	mp2an 692	. . . . . 6 ⊢ ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
31		enreceq 10957	. . . . . 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))))))
32	6, 4, 24, 30, 31	mp4an 693	. . . . 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)))))
33	18, 32	mpbir 231	. . . 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
34	8, 33	eqtr4i 2757	. . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
35	3, 34	eqtr4i 2757	. 2 ⊢ 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
36	2, 35	eqtr4i 2757	1 ⊢ (-1R ·R -1R) = 1R

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ⟨cop 4579  (class class class)co 7346  [cec 8620  Pcnp 10750  1_Pc1p 10751   +_P cpp 10752   ·_P cmp 10753   ~_R cer 10755  1_Rc1r 10758  -1_Rcm1r 10759   ·_R cmr 10761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-ec 8624  df-qs 8628  df-ni 10763  df-pli 10764  df-mi 10765  df-lti 10766  df-plpq 10799  df-mpq 10800  df-ltpq 10801  df-enq 10802  df-nq 10803  df-erq 10804  df-plq 10805  df-mq 10806  df-1nq 10807  df-rq 10808  df-ltnq 10809  df-np 10872  df-1p 10873  df-plp 10874  df-mp 10875  df-ltp 10876  df-enr 10946  df-nr 10947  df-mr 10949  df-1r 10952  df-m1r 10953

This theorem is referenced by:  sqgt0sr  10997