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Theorem m1m1sr 10519
 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 10488 . . 3 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21, 1oveq12i 7154 . 2 (-1R ·R -1R) = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
3 df-1r 10487 . . 3 1R = [⟨(1P +P 1P), 1P⟩] ~R
4 1pr 10441 . . . . 5 1PP
5 addclpr 10444 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
64, 4, 5mp2an 691 . . . . 5 (1P +P 1P) ∈ P
7 mulsrpr 10502 . . . . 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 692 . . . 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 10448 . . . . . 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 10455 . . . . . . . . 9 (1PP → (1P ·P 1P) = 1P)
114, 10ax-mp 5 . . . . . . . 8 (1P ·P 1P) = 1P
12 distrpr 10454 . . . . . . . . 9 ((1P +P 1P) ·P (1P +P 1P)) = (((1P +P 1P) ·P 1P) +P ((1P +P 1P) ·P 1P))
13 mulcompr 10449 . . . . . . . . . 10 (1P ·P (1P +P 1P)) = ((1P +P 1P) ·P 1P)
1413oveq1i 7152 . . . . . . . . 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 2824 . . . . . . . 8 ((1P +P 1P) ·P (1P +P 1P)) = ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P))
1611, 15oveq12i 7154 . . . . . . 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 7153 . . . . . 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 2824 . . . . 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 10446 . . . . . . . 8 ((1PP ∧ 1PP) → (1P ·P 1P) ∈ P)
204, 4, 19mp2an 691 . . . . . . 7 (1P ·P 1P) ∈ P
21 mulclpr 10446 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ (1P +P 1P) ∈ P) → ((1P +P 1P) ·P (1P +P 1P)) ∈ P)
226, 6, 21mp2an 691 . . . . . . 7 ((1P +P 1P) ·P (1P +P 1P)) ∈ P
23 addclpr 10444 . . . . . . 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 691 . . . . . 6 ((1P ·P 1P) +P ((1P +P 1P) ·P (1P +P 1P))) ∈ P
25 mulclpr 10446 . . . . . . . 8 ((1PP ∧ (1P +P 1P) ∈ P) → (1P ·P (1P +P 1P)) ∈ P)
264, 6, 25mp2an 691 . . . . . . 7 (1P ·P (1P +P 1P)) ∈ P
27 mulclpr 10446 . . . . . . . 8 (((1P +P 1P) ∈ P ∧ 1PP) → ((1P +P 1P) ·P 1P) ∈ P)
286, 4, 27mp2an 691 . . . . . . 7 ((1P +P 1P) ·P 1P) ∈ P
29 addclpr 10444 . . . . . . 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 691 . . . . . 6 ((1P ·P (1P +P 1P)) +P ((1P +P 1P) ·P 1P)) ∈ P
31 enreceq 10492 . . . . . 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 692 . . . . 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 234 . . . 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 2824 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R
353, 34eqtr4i 2824 . 2 1R = ([⟨1P, (1P +P 1P)⟩] ~R ·R [⟨1P, (1P +P 1P)⟩] ~R )
362, 35eqtr4i 2824 1 (-1R ·R -1R) = 1R
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ⟨cop 4533  (class class class)co 7142  [cec 8285  Pcnp 10285  1Pc1p 10286   +P cpp 10287   ·P cmp 10288   ~R cer 10290  1Rc1r 10293  -1Rcm1r 10294   ·R cmr 10296 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-inf2 9103 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7571  df-1st 7681  df-2nd 7682  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-omul 8105  df-er 8287  df-ec 8289  df-qs 8293  df-ni 10298  df-pli 10299  df-mi 10300  df-lti 10301  df-plpq 10334  df-mpq 10335  df-ltpq 10336  df-enq 10337  df-nq 10338  df-erq 10339  df-plq 10340  df-mq 10341  df-1nq 10342  df-rq 10343  df-ltnq 10344  df-np 10407  df-1p 10408  df-plp 10409  df-mp 10410  df-ltp 10411  df-enr 10481  df-nr 10482  df-mr 10484  df-1r 10487  df-m1r 10488 This theorem is referenced by:  sqgt0sr  10532
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