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Theorem m1m1sr 11131
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 11100 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 7443 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 11099 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 11053 . . . . 5 1PP
5 addclpr 11056 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
64, 4, 5mp2an 692 . . . . 5 (1P +P 1P) ∈ P
7 mulsrpr 11114 . . . . 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 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 11060 . . . . . 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 11067 . . . . . . . . 9 (1PP → (1P ·P 1P) = 1P)
114, 10ax-mp 5 . . . . . . . 8 (1P ·P 1P) = 1P
12 distrpr 11066 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
13 mulcompr 11061 . . . . . . . . . 10 (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
1413oveq1i 7441 . . . . . . . . 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 2766 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
1611, 15oveq12i 7443 . . . . . . 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 7442 . . . . . 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 2766 . . . . 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 11058 . . . . . . . 8 ((1PP ∧ 1PP) → (1P ·P 1P) ∈ P)
204, 4, 19mp2an 692 . . . . . . 7 (1P ·P 1P) ∈ P
21 mulclpr 11058 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
226, 6, 21mp2an 692 . . . . . . 7 ((1P +P 1P) ·P (1P +P 1P)) ∈ P
23 addclpr 11056 . . . . . . 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 692 . . . . . 6 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
25 mulclpr 11058 . . . . . . . 8 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
264, 6, 25mp2an 692 . . . . . . 7 (1P ·P (1P +P 1P)) ∈ P
27 mulclpr 11058 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) ·P 1P) ∈ P)
286, 4, 27mp2an 692 . . . . . . 7 ((1P +P 1P) ·P 1P) ∈ P
29 addclpr 11056 . . . . . . 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 692 . . . . . 6 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
31 enreceq 11104 . . . . . 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 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)))))
3318, 32mpbir 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
348, 33eqtr4i 2766 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
353, 34eqtr4i 2766 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
362, 35eqtr4i 2766 1 (-1R ·R -1R) = 1R
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  cop 4637  (class class class)co 7431  [cec 8742  Pcnp 10897  1Pc1p 10898   +P cpp 10899   ·P cmp 10900   ~R cer 10902  1Rc1r 10905  -1Rcm1r 10906   ·R cmr 10908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-1p 11020  df-plp 11021  df-mp 11022  df-ltp 11023  df-enr 11093  df-nr 11094  df-mr 11096  df-1r 11099  df-m1r 11100
This theorem is referenced by:  sqgt0sr  11144
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