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Theorem m1m1sr 9952
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
StepHypRef Expression
1 df-m1r 9922 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 6702 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 9921 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 9875 . . . . 5 1PP
5 addclpr 9878 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
64, 4, 5mp2an 708 . . . . 5 (1P +P 1P) ∈ P
7 mulsrpr 9935 . . . . 5 (((1PP ∧ (1P +P 1P) ∈ P) ∧ (1PP ∧ (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 )
84, 6, 4, 6, 7mp4an 709 . . . 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 9882 . . . . . 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 9889 . . . . . . . . 9 (1PP → (1P ·P 1P) = 1P)
114, 10ax-mp 5 . . . . . . . 8 (1P ·P 1P) = 1P
12 distrpr 9888 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
13 mulcompr 9883 . . . . . . . . . 10 (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
1413oveq1i 6700 . . . . . . . . 9 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
1512, 14eqtr4i 2676 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
1611, 15oveq12i 6702 . . . . . . 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)))
1716oveq2i 6701 . . . . . 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))))
189, 17eqtr4i 2676 . . . . 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 9880 . . . . . . . 8 ((1PP ∧ 1PP) → (1P ·P 1P) ∈ P)
204, 4, 19mp2an 708 . . . . . . 7 (1P ·P 1P) ∈ P
21 mulclpr 9880 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
226, 6, 21mp2an 708 . . . . . . 7 ((1P +P 1P) ·P (1P +P 1P)) ∈ P
23 addclpr 9878 . . . . . . 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)
2420, 22, 23mp2an 708 . . . . . 6 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
25 mulclpr 9880 . . . . . . . 8 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
264, 6, 25mp2an 708 . . . . . . 7 (1P ·P (1P +P 1P)) ∈ P
27 mulclpr 9880 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) ·P 1P) ∈ P)
286, 4, 27mp2an 708 . . . . . . 7 ((1P +P 1P) ·P 1P) ∈ P
29 addclpr 9878 . . . . . . 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)
3026, 28, 29mp2an 708 . . . . . 6 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
31 enreceq 9925 . . . . . 6 ((((1P +P 1P) ∈ P ∧ 1PP) ∧ (((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))))))
326, 4, 24, 30, 31mp4an 709 . . . . 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)))))
3318, 32mpbir 221 . . . 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
348, 33eqtr4i 2676 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
353, 34eqtr4i 2676 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
362, 35eqtr4i 2676 1 (-1R ·R -1R) = 1R
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  cop 4216  (class class class)co 6690  [cec 7785  Pcnp 9719  1Pc1p 9720   +P cpp 9721   ·P cmp 9722   ~R cer 9724  1Rc1r 9727  -1Rcm1r 9728   ·R cmr 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-mr 9918  df-1r 9921  df-m1r 9922
This theorem is referenced by:  sqgt0sr  9965
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