Theorem nq0m0r 7659

Description: Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)

Assertion

Ref	Expression
nq0m0r	⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0)

Proof of Theorem nq0m0r

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nq0nn 7645	. 2 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2		df-0nq0 7629	. . . . . 6 ⊢ 0Q0 = [⟨∅, 1o⟩] ~Q0
3		oveq12 6019	. . . . . 6 ⊢ ((0Q0 = [⟨∅, 1o⟩] ~Q0 ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = ([⟨∅, 1o⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
4	2, 3	mpan 424	. . . . 5 ⊢ (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 → (0Q0 ·Q0 𝐴) = ([⟨∅, 1o⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
5		peano1 4687	. . . . . 6 ⊢ ∅ ∈ ω
6		1pi 7518	. . . . . 6 ⊢ 1o ∈ N
7		mulnnnq0 7653	. . . . . 6 ⊢ (((∅ ∈ ω ∧ 1o ∈ N) ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) → ([⟨∅, 1o⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 )
8	5, 6, 7	mpanl12 436	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ([⟨∅, 1o⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 )
9	4, 8	sylan9eqr 2284	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 )
10		nnm0r 6638	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∅ ·o 𝑤) = ∅)
11	10	oveq1d 6025	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ·o 𝑤) ·o 1o) = (∅ ·o 1o))
12		1onn 6679	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		nnm0r 6638	. . . . . . . . . . 11 ⊢ (1o ∈ ω → (∅ ·o 1o) = ∅)
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (∅ ·o 1o) = ∅
15	11, 14	eqtrdi 2278	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((∅ ·o 𝑤) ·o 1o) = ∅)
16	15	adantr 276	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·o 𝑤) ·o 1o) = ∅)
17		mulpiord 7520	. . . . . . . . . . . 12 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·N 𝑣) = (1o ·o 𝑣))
18		mulclpi 7531	. . . . . . . . . . . 12 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·N 𝑣) ∈ N)
19	17, 18	eqeltrrd 2307	. . . . . . . . . . 11 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·o 𝑣) ∈ N)
20	6, 19	mpan 424	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → (1o ·o 𝑣) ∈ N)
21		pinn 7512	. . . . . . . . . 10 ⊢ ((1o ·o 𝑣) ∈ N → (1o ·o 𝑣) ∈ ω)
22		nnm0 6634	. . . . . . . . . 10 ⊢ ((1o ·o 𝑣) ∈ ω → ((1o ·o 𝑣) ·o ∅) = ∅)
23	20, 21, 22	3syl 17	. . . . . . . . 9 ⊢ (𝑣 ∈ N → ((1o ·o 𝑣) ·o ∅) = ∅)
24	23	adantl 277	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((1o ·o 𝑣) ·o ∅) = ∅)
25	16, 24	eqtr4d 2265	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·o 𝑤) ·o 1o) = ((1o ·o 𝑣) ·o ∅))
26	10, 5	eqeltrdi 2320	. . . . . . . 8 ⊢ (𝑤 ∈ ω → (∅ ·o 𝑤) ∈ ω)
27		enq0eceq 7640	. . . . . . . . 9 ⊢ ((((∅ ·o 𝑤) ∈ ω ∧ (1o ·o 𝑣) ∈ N) ∧ (∅ ∈ ω ∧ 1o ∈ N)) → ([⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = [⟨∅, 1o⟩] ~Q0 ↔ ((∅ ·o 𝑤) ·o 1o) = ((1o ·o 𝑣) ·o ∅)))
28	5, 6, 27	mpanr12 439	. . . . . . . 8 ⊢ (((∅ ·o 𝑤) ∈ ω ∧ (1o ·o 𝑣) ∈ N) → ([⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = [⟨∅, 1o⟩] ~Q0 ↔ ((∅ ·o 𝑤) ·o 1o) = ((1o ·o 𝑣) ·o ∅)))
29	26, 20, 28	syl2an 289	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ([⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = [⟨∅, 1o⟩] ~Q0 ↔ ((∅ ·o 𝑤) ·o 1o) = ((1o ·o 𝑣) ·o ∅)))
30	25, 29	mpbird 167	. . . . . 6 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = [⟨∅, 1o⟩] ~Q0 )
31	30, 2	eqtr4di 2280	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = 0Q0)
32	31	adantr 276	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 = 0Q0)
33	9, 32	eqtrd 2262	. . 3 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
34	33	exlimivv 1943	. 2 ⊢ (∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
35	1, 34	syl 14	1 ⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∅c0 3491  ⟨cop 3669  ωcom 4683  (class class class)co 6010  1_oc1o 6566   ·_o comu 6571  [cec 6691  Ncnpi 7475   ·_N cmi 7477   ~_Q0 ceq0 7489  Q₀cnq0 7490  0_Q0c0q0 7491   ·_Q0 cmq0 7493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-mi 7509  df-enq0 7627  df-nq0 7628  df-0nq0 7629  df-mq0 7631

This theorem is referenced by:  prarloclem5  7703