Theorem nq0m0r 7676

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 7662	. 2 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2		df-0nq0 7646	. . . . . 6 ⊢ 0Q0 = [⟨∅, 1o⟩] ~Q0
3		oveq12 6027	. . . . . 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 4692	. . . . . 6 ⊢ ∅ ∈ ω
6		1pi 7535	. . . . . 6 ⊢ 1o ∈ N
7		mulnnnq0 7670	. . . . . 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 2286	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·o 𝑤), (1o ·o 𝑣)⟩] ~Q0 )
10		nnm0r 6647	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∅ ·o 𝑤) = ∅)
11	10	oveq1d 6033	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ·o 𝑤) ·o 1o) = (∅ ·o 1o))
12		1onn 6688	. . . . . . . . . . 11 ⊢ 1o ∈ ω
13		nnm0r 6647	. . . . . . . . . . 11 ⊢ (1o ∈ ω → (∅ ·o 1o) = ∅)
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ (∅ ·o 1o) = ∅
15	11, 14	eqtrdi 2280	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((∅ ·o 𝑤) ·o 1o) = ∅)
16	15	adantr 276	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·o 𝑤) ·o 1o) = ∅)
17		mulpiord 7537	. . . . . . . . . . . 12 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·N 𝑣) = (1o ·o 𝑣))
18		mulclpi 7548	. . . . . . . . . . . 12 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·N 𝑣) ∈ N)
19	17, 18	eqeltrrd 2309	. . . . . . . . . . 11 ⊢ ((1o ∈ N ∧ 𝑣 ∈ N) → (1o ·o 𝑣) ∈ N)
20	6, 19	mpan 424	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → (1o ·o 𝑣) ∈ N)
21		pinn 7529	. . . . . . . . . 10 ⊢ ((1o ·o 𝑣) ∈ N → (1o ·o 𝑣) ∈ ω)
22		nnm0 6643	. . . . . . . . . 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 2267	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·o 𝑤) ·o 1o) = ((1o ·o 𝑣) ·o ∅))
26	10, 5	eqeltrdi 2322	. . . . . . . 8 ⊢ (𝑤 ∈ ω → (∅ ·o 𝑤) ∈ ω)
27		enq0eceq 7657	. . . . . . . . 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 2282	. . . . 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 2264	. . 3 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
34	33	exlimivv 1945	. 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 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494  ⟨cop 3672  ωcom 4688  (class class class)co 6018  1_oc1o 6575   ·_o comu 6580  [cec 6700  Ncnpi 7492   ·_N cmi 7494   ~_Q0 ceq0 7506  Q₀cnq0 7507  0_Q0c0q0 7508   ·_Q0 cmq0 7510

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-mi 7526  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-mq0 7648

This theorem is referenced by:  prarloclem5  7720