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Theorem 00sr 7008
Description: A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
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 6966 . 2 R = ((P × P) / ~R )
2 oveq1 5550 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = (𝐴 ·R 0R))
32eqeq1d 2090 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R ↔ (𝐴 ·R 0R) = 0R))
4 1pr 6806 . . . . 5 1PP
5 mulsrpr 6985 . . . . 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 430 . . . 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 6824 . . . . . . . . . 10 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
84, 7mpan2 416 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
9 mulclpr 6824 . . . . . . . . . 10 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
104, 9mpan2 416 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
11 addclpr 6789 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
128, 10, 11syl2an 283 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
1312, 12anim12i 331 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P))
14 eqid 2082 . . . . . . . 8 (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)
15 enreceq 6975 . . . . . . . 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 166 . . . . . . 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 277 . . . . . 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 430 . . . . 5 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
1918anidms 389 . . . 4 ((𝑥P𝑦P) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
206, 19eqtrd 2114 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21 df-0r 6970 . . . 4 0R = [⟨1P, 1P⟩] ~R
2221oveq2i 5554 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R )
2320, 22, 213eqtr4g 2139 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R)
241, 3, 23ecoptocl 6259 1 (𝐴R → (𝐴 ·R 0R) = 0R)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  cop 3409  (class class class)co 5543  [cec 6170  Pcnp 6543  1Pc1p 6544   +P cpp 6545   ·P cmp 6546   ~R cer 6548  Rcnr 6549  0Rc0r 6550   ·R cmr 6554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-enr 6965  df-nr 6966  df-mr 6968  df-0r 6970
This theorem is referenced by:  pn0sr  7010  mulresr  7068  axi2m1  7103  axcnre  7109
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