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Theorem 00sr 10513
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 10470 . 2 R = ((P × P) / ~R )
2 oveq1 7155 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = (𝐴 ·R 0R))
32eqeq1d 2821 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R ↔ (𝐴 ·R 0R) = 0R))
4 1pr 10429 . . . . 5 1PP
5 mulsrpr 10490 . . . . 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 703 . . . 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 10434 . . . . . . . . . 10 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
84, 7mpan2 689 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
9 mulclpr 10434 . . . . . . . . . 10 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
104, 9mpan2 689 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
11 addclpr 10432 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
128, 10, 11syl2an 597 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
1312, 12anim12i 614 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P))
14 eqid 2819 . . . . . . . 8 (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)
15 enreceq 10480 . . . . . . . 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 260 . . . . . . 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 582 . . . . . 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 703 . . . . 5 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
1918anidms 569 . . . 4 ((𝑥P𝑦P) → [⟨((𝑥 ·P 1P) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
206, 19eqtrd 2854 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21 df-0r 10474 . . . 4 0R = [⟨1P, 1P⟩] ~R
2221oveq2i 7159 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R )
2320, 22, 213eqtr4g 2879 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R)
241, 3, 23ecoptocl 8379 1 (𝐴R → (𝐴 ·R 0R) = 0R)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  cop 4565  (class class class)co 7148  [cec 8279  Pcnp 10273  1Pc1p 10274   +P cpp 10275   ·P cmp 10276   ~R cer 10278  Rcnr 10279  0Rc0r 10280   ·R cmr 10284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-omul 8099  df-er 8281  df-ec 8283  df-qs 8287  df-ni 10286  df-pli 10287  df-mi 10288  df-lti 10289  df-plpq 10322  df-mpq 10323  df-ltpq 10324  df-enq 10325  df-nq 10326  df-erq 10327  df-plq 10328  df-mq 10329  df-1nq 10330  df-rq 10331  df-ltnq 10332  df-np 10395  df-1p 10396  df-plp 10397  df-mp 10398  df-ltp 10399  df-enr 10469  df-nr 10470  df-mr 10472  df-0r 10474
This theorem is referenced by:  pn0sr  10515  mulresr  10553  axi2m1  10573  axcnre  10578
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