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Theorem 00sr 9958
Description: A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
00sr (𝐴R → (𝐴 ·R 0R) = 0R)

Proof of Theorem 00sr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9916 . 2 R = ((P × P) / ~R )
2 oveq1 6697 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = (𝐴 ·R 0R))
32eqeq1d 2653 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R ↔ (𝐴 ·R 0R) = 0R))
4 1pr 9875 . . . . 5 1PP
5 mulsrpr 9935 . . . . 5 (((𝑥P𝑦P) ∧ (1PP ∧ 1PP)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R )
64, 4, 5mpanr12 721 . . . 4 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R )
7 mulclpr 9880 . . . . . . . . . 10 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
84, 7mpan2 707 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
9 mulclpr 9880 . . . . . . . . . 10 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
104, 9mpan2 707 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
11 addclpr 9878 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
128, 10, 11syl2an 493 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
1312, 12anim12i 589 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P))
14 eqid 2651 . . . . . . . 8 (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)
15 enreceq 9925 . . . . . . . 8 (((((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P) ∧ (1PP ∧ 1PP)) → ([⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R ↔ (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)))
1614, 15mpbiri 248 . . . . . . 7 (((((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P) ∧ (1PP ∧ 1PP)) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
1713, 16sylan 487 . . . . . 6 ((((𝑥P𝑦P) ∧ (𝑥P𝑦P)) ∧ (1PP ∧ 1PP)) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
184, 4, 17mpanr12 721 . . . . 5 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
1918anidms 678 . . . 4 ((𝑥P𝑦P) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
206, 19eqtrd 2685 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21 df-0r 9920 . . . 4 0R = [⟨1P, 1P⟩] ~R
2221oveq2i 6701 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R )
2320, 22, 213eqtr4g 2710 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R)
241, 3, 23ecoptocl 7880 1 (𝐴R → (𝐴 ·R 0R) = 0R)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = 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  Rcnr 9725  0Rc0r 9726   ·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-0r 9920
This theorem is referenced by:  pn0sr  9960  mulresr  9998  axi2m1  10018  axcnre  10023
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