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Theorem m1m1sr 9793
 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 9763 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 6561 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 9762 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 9716 . . . . 5 1PP
5 addclpr 9719 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
64, 4, 5mp2an 704 . . . . 5 (1P +P 1P) ∈ P
7 mulsrpr 9776 . . . . 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 705 . . . 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 9723 . . . . . 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 9730 . . . . . . . . 9 (1PP → (1P ·P 1P) = 1P)
114, 10ax-mp 5 . . . . . . . 8 (1P ·P 1P) = 1P
12 distrpr 9729 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
13 mulcompr 9724 . . . . . . . . . 10 (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
1413oveq1i 6559 . . . . . . . . 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 2635 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
1611, 15oveq12i 6561 . . . . . . 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 6560 . . . . . 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 2635 . . . . 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 9721 . . . . . . . 8 ((1PP ∧ 1PP) → (1P ·P 1P) ∈ P)
204, 4, 19mp2an 704 . . . . . . 7 (1P ·P 1P) ∈ P
21 mulclpr 9721 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
226, 6, 21mp2an 704 . . . . . . 7 ((1P +P 1P) ·P (1P +P 1P)) ∈ P
23 addclpr 9719 . . . . . . 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 704 . . . . . 6 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
25 mulclpr 9721 . . . . . . . 8 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
264, 6, 25mp2an 704 . . . . . . 7 (1P ·P (1P +P 1P)) ∈ P
27 mulclpr 9721 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) ·P 1P) ∈ P)
286, 4, 27mp2an 704 . . . . . . 7 ((1P +P 1P) ·P 1P) ∈ P
29 addclpr 9719 . . . . . . 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 704 . . . . . 6 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
31 enreceq 9766 . . . . . 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 705 . . . . 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 220 . . . 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 2635 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
353, 34eqtr4i 2635 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
362, 35eqtr4i 2635 1 (-1R ·R -1R) = 1R
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  (class class class)co 6549  [cec 7627  Pcnp 9560  1Pc1p 9561   +P cpp 9562   ·P cmp 9563   ~R cer 9565  1Rc1r 9568  -1Rcm1r 9569   ·R cmr 9571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-1p 9683  df-plp 9684  df-mp 9685  df-ltp 9686  df-enr 9756  df-nr 9757  df-mr 9759  df-1r 9762  df-m1r 9763 This theorem is referenced by:  sqgt0sr  9806
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