Theorem nq0m0r 6646

Description: Multiplication with zero for non-negative 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 6632	. 2 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2		df-0nq0 6616	. . . . . 6 ⊢ 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
3		oveq12 5541	. . . . . 6 ⊢ ((0Q0 = [⟨∅, 1𝑜⟩] ~Q0 ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
4	2, 3	mpan 414	. . . . 5 ⊢ (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
5		peano1 4335	. . . . . 6 ⊢ ∅ ∈ ω
6		1pi 6505	. . . . . 6 ⊢ 1𝑜 ∈ N
7		mulnnnq0 6640	. . . . . 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 426	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
9	4, 8	sylan9eqr 2135	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
10		nnm0r 6081	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) = ∅)
11	10	oveq1d 5547	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = (∅ ·𝑜 1𝑜))
12		1onn 6116	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ ω
13		nnm0r 6081	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ ω → (∅ ·𝑜 1𝑜) = ∅)
14	12, 13	ax-mp 7	. . . . . . . . . 10 ⊢ (∅ ·𝑜 1𝑜) = ∅
15	11, 14	syl6eq 2129	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
16	15	adantr 270	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
17		mulpiord 6507	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) = (1𝑜 ·𝑜 𝑣))
18		mulclpi 6518	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) ∈ N)
19	17, 18	eqeltrrd 2156	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·𝑜 𝑣) ∈ N)
20	6, 19	mpan 414	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → (1𝑜 ·𝑜 𝑣) ∈ N)
21		pinn 6499	. . . . . . . . . 10 ⊢ ((1𝑜 ·𝑜 𝑣) ∈ N → (1𝑜 ·𝑜 𝑣) ∈ ω)
22		nnm0 6077	. . . . . . . . . 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 271	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅) = ∅)
25	16, 24	eqtr4d 2116	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅))
26	10, 5	syl6eqel 2169	. . . . . . . 8 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) ∈ ω)
27		enq0eceq 6627	. . . . . . . . 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 429	. . . . . . . 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 283	. . . . . . 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 165	. . . . . 6 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 )
31	30, 2	syl6eqr 2131	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
32	31	adantr 270	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
33	9, 32	eqtrd 2113	. . 3 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
34	33	exlimivv 1817	. 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 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∅c0 3251  ⟨cop 3401  ωcom 4331  (class class class)co 5532  1_𝑜c1o 6017   ·_𝑜 comu 6022  [cec 6127  Ncnpi 6462   ·_N cmi 6464   ~_Q0 ceq0 6476  Q₀cnq0 6477  0_Q0c0q0 6478   ·_Q0 cmq0 6480

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-mq0 6618

This theorem is referenced by:  prarloclem5  6690