Theorem nq0m0r 6554

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 6540	. 2 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2		df-0nq0 6524	. . . . . 6 ⊢ 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
3		oveq12 5521	. . . . . 6 ⊢ ((0Q0 = [⟨∅, 1𝑜⟩] ~Q0 ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
4	2, 3	mpan 400	. . . . 5 ⊢ (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
5		peano1 4317	. . . . . 6 ⊢ ∅ ∈ ω
6		1pi 6413	. . . . . 6 ⊢ 1𝑜 ∈ N
7		mulnnnq0 6548	. . . . . 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 412	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
9	4, 8	sylan9eqr 2094	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
10		nnm0r 6058	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) = ∅)
11	10	oveq1d 5527	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = (∅ ·𝑜 1𝑜))
12		1onn 6093	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ ω
13		nnm0r 6058	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ ω → (∅ ·𝑜 1𝑜) = ∅)
14	12, 13	ax-mp 7	. . . . . . . . . 10 ⊢ (∅ ·𝑜 1𝑜) = ∅
15	11, 14	syl6eq 2088	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
16	15	adantr 261	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
17		mulpiord 6415	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) = (1𝑜 ·𝑜 𝑣))
18		mulclpi 6426	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·N 𝑣) ∈ N)
19	17, 18	eqeltrrd 2115	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ N ∧ 𝑣 ∈ N) → (1𝑜 ·𝑜 𝑣) ∈ N)
20	6, 19	mpan 400	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → (1𝑜 ·𝑜 𝑣) ∈ N)
21		pinn 6407	. . . . . . . . . 10 ⊢ ((1𝑜 ·𝑜 𝑣) ∈ N → (1𝑜 ·𝑜 𝑣) ∈ ω)
22		nnm0 6054	. . . . . . . . . 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 262	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅) = ∅)
25	16, 24	eqtr4d 2075	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅))
26	10, 5	syl6eqel 2128	. . . . . . . 8 ⊢ (𝑤 ∈ ω → (∅ ·𝑜 𝑤) ∈ ω)
27		enq0eceq 6535	. . . . . . . . 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 415	. . . . . . . 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 273	. . . . . . 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 156	. . . . . 6 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 )
31	30, 2	syl6eqr 2090	. . . . 5 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
32	31	adantr 261	. . . 4 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
33	9, 32	eqtrd 2072	. . 3 ⊢ (((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
34	33	exlimivv 1776	. 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 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∅c0 3224  ⟨cop 3378  ωcom 4313  (class class class)co 5512  1_𝑜c1o 5994   ·_𝑜 comu 5999  [cec 6104  Ncnpi 6370   ·_N cmi 6372   ~_Q0 ceq0 6384  Q₀cnq0 6385  0_Q0c0q0 6386   ·_Q0 cmq0 6388

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-mq0 6526

This theorem is referenced by:  prarloclem5  6598