Theorem nq0m0r 7013

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 6999	. 2 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2		df-0nq0 6983	. . . . . 6 ⊢ 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
3		oveq12 5661	. . . . . 6 ⊢ ((0Q0 = [⟨∅, 1𝑜⟩] ~Q0 ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
4	2, 3	mpan 415	. . . . 5 ⊢ (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
5		peano1 4409	. . . . . 6 ⊢ ∅ ∈ ω
6		1pi 6872	. . . . . 6 ⊢ 1𝑜 ∈ N
7		mulnnnq0 7007	. . . . . 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 427	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
9	4, 8	sylan9eqr 2142	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
10		nnm0r 6240	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) = ∅)
11	10	oveq1d 5667	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = (∅ ·𝑜 1𝑜))
12		1onn 6277	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ ω
13		nnm0r 6240	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ ω → (∅ ·𝑜 1𝑜) = ∅)
14	12, 13	ax-mp 7	. . . . . . . . . 10 ⊢ (∅ ·𝑜 1𝑜) = ∅
15	11, 14	syl6eq 2136	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
16	15	adantr 270	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
17		mulpiord 6874	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) = (1𝑜 ·𝑜 𝑣))
18		mulclpi 6885	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) ∈ N)
19	17, 18	eqeltrrd 2165	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·𝑜 𝑣) ∈ N)
20	6, 19	mpan 415	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → (1𝑜 ·𝑜 𝑣) ∈ N)
21		pinn 6866	. . . . . . . . . 10 ⊢ ((1𝑜 ·𝑜 𝑣) ∈ N → (1𝑜 ·𝑜 𝑣) ∈ ω)
22		nnm0 6236	. . . . . . . . . 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 2123	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅))
26	10, 5	syl6eqel 2178	. . . . . . . 8 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) ∈ ω)
27		enq0eceq 6994	. . . . . . . . 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 430	. . . . . . . 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 2138	. . . . 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 2120	. . 3 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
34	33	exlimivv 1824	. 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 1289  ∃wex 1426   ∈ wcel 1438  ∅c0 3286  ⟨cop 3449  ωcom 4405  (class class class)co 5652  1_𝑜c1o 6174   ·_𝑜 comu 6179  [cec 6288  Ncnpi 6829   ·_N cmi 6831   ~_Q0 ceq0 6843  Q₀cnq0 6844  0_Q0c0q0 6845   ·_Q0 cmq0 6847

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-mi 6863  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-mq0 6985

This theorem is referenced by:  prarloclem5  7057