Theorem 00sr 4002

Description: A signed real times 0 is 0.

Assertion

Ref	Expression
00sr	⊢ (A ∈ R → (A ·R 0R) = 0R)

Proof of Theorem 00sr

Step	Hyp	Ref	Expression
1		df-nr 3961	. 2 ⊢ R = ((P × P) / ~R )
2		opreq1 3006	. . 3 ⊢ ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R ·R 0R) = (A ·R 0R))
3	2	cleq1d 1109	. 2 ⊢ ([⟨x, y⟩] ~R = A → (([⟨x, y⟩] ~R ·R 0R) = 0R ↔ (A ·R 0R) = 0R))
4		1pr 3911	. . . . . 6 ⊢ 1P ∈ P
5	4, 4	pm3.2i 234	. . . . 5 ⊢ (1P ∈ P ∧ 1P ∈ P)
6		mulsrpr 3979	. . . . 5 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R )
7	5, 6	mpan2 519	. . . 4 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R )
8		cleqid 1102	. . . . . . . 8 ⊢ (((x ·P 1P) +P (y ·P 1P)) +P 1P) = (((x ·P 1P) +P (y ·P 1P)) +P 1P)
9		enreceq 3971	. . . . . . . 8 ⊢ (((((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → ([⟨((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)))
10	8, 9	mpbiri 169	. . . . . . 7 ⊢ (((((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
11		addclpr 3914	. . . . . . . . 9 ⊢ (((x ·P 1P) ∈ P ∧ (y ·P 1P) ∈ P) → ((x ·P 1P) +P (y ·P 1P)) ∈ P)
12		mulclpr 3916	. . . . . . . . . 10 ⊢ ((x ∈ P ∧ 1P ∈ P) → (x ·P 1P) ∈ P)
13	4, 12	mpan2 519	. . . . . . . . 9 ⊢ (x ∈ P → (x ·P 1P) ∈ P)
14		mulclpr 3916	. . . . . . . . . 10 ⊢ ((y ∈ P ∧ 1P ∈ P) → (y ·P 1P) ∈ P)
15	4, 14	mpan2 519	. . . . . . . . 9 ⊢ (y ∈ P → (y ·P 1P) ∈ P)
16	11, 13, 15	syl2an 349	. . . . . . . 8 ⊢ ((x ∈ P ∧ y ∈ P) → ((x ·P 1P) +P (y ·P 1P)) ∈ P)
17	16, 16	anim12i 268	. . . . . . 7 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → (((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P))
18	10, 17	sylan 343	. . . . . 6 ⊢ ((((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) ∧ (1P ∈ P ∧ 1P ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
19	5, 18	mpan2 519	. . . . 5 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
20	19	anidms 332	. . . 4 ⊢ ((x ∈ P ∧ y ∈ P) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
21	7, 20	eqtrd 1128	. . 3 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
22		df-0r 3965	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
23	22	opreq2i 3010	. . 3 ⊢ ([⟨x, y⟩] ~R ·R 0R) = ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R )
24	21, 23, 22	3eqtr4g 1147	. 2 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R 0R) = 0R)
25	1, 3, 24	ecoptocl 3239	1 ⊢ (A ∈ R → (A ·R 0R) = 0R)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  [cec 3198  Pcnp 3779  1_Pc1p 3780   +_P cpp 3781   ·_P cmp 3782   ~_R cer 3786  Rcnr 3787  0_Rc0r 3788   ·_R cmr 3792

This theorem is referenced by:  pn0sr 4004  ssgt0sr 4011  mulresr 4051  ax1id 4077  axi2m1 4082  axcnre 4087

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-mpr 3959  df-enr 3960  df-nr 3961  df-mr 3963  df-0r 3965

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