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Theorem nq0m0r 6554
 Description: Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
Assertion
Ref Expression
nq0m0r (𝐴Q0 → (0Q0 ·Q0 𝐴) = 0Q0)

Proof of Theorem nq0m0r
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nq0nn 6540 . 2 (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
2 df-0nq0 6524 . . . . . 6 0Q0 = [⟨∅, 1𝑜⟩] ~Q0
3 oveq12 5521 . . . . . 6 ((0Q0 = [⟨∅, 1𝑜⟩] ~Q0𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
42, 3mpan 400 . . . . 5 (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 → (0Q0 ·Q0 𝐴) = ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ))
5 peano1 4317 . . . . . 6 ∅ ∈ ω
6 1pi 6413 . . . . . 6 1𝑜N
7 mulnnnq0 6548 . . . . . 6 (((∅ ∈ ω ∧ 1𝑜N) ∧ (𝑤 ∈ ω ∧ 𝑣N)) → ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
85, 6, 7mpanl12 412 . . . . 5 ((𝑤 ∈ ω ∧ 𝑣N) → ([⟨∅, 1𝑜⟩] ~Q0 ·Q0 [⟨𝑤, 𝑣⟩] ~Q0 ) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
94, 8sylan9eqr 2094 . . . 4 (((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 )
10 nnm0r 6058 . . . . . . . . . . 11 (𝑤 ∈ ω → (∅ ·𝑜 𝑤) = ∅)
1110oveq1d 5527 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = (∅ ·𝑜 1𝑜))
12 1onn 6093 . . . . . . . . . . 11 1𝑜 ∈ ω
13 nnm0r 6058 . . . . . . . . . . 11 (1𝑜 ∈ ω → (∅ ·𝑜 1𝑜) = ∅)
1412, 13ax-mp 7 . . . . . . . . . 10 (∅ ·𝑜 1𝑜) = ∅
1511, 14syl6eq 2088 . . . . . . . . 9 (𝑤 ∈ ω → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
1615adantr 261 . . . . . . . 8 ((𝑤 ∈ ω ∧ 𝑣N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ∅)
17 mulpiord 6415 . . . . . . . . . . . 12 ((1𝑜N𝑣N) → (1𝑜 ·N 𝑣) = (1𝑜 ·𝑜 𝑣))
18 mulclpi 6426 . . . . . . . . . . . 12 ((1𝑜N𝑣N) → (1𝑜 ·N 𝑣) ∈ N)
1917, 18eqeltrrd 2115 . . . . . . . . . . 11 ((1𝑜N𝑣N) → (1𝑜 ·𝑜 𝑣) ∈ N)
206, 19mpan 400 . . . . . . . . . 10 (𝑣N → (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 ∅) = ∅)
2320, 21, 223syl 17 . . . . . . . . 9 (𝑣N → ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅) = ∅)
2423adantl 262 . . . . . . . 8 ((𝑤 ∈ ω ∧ 𝑣N) → ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅) = ∅)
2516, 24eqtr4d 2075 . . . . . . 7 ((𝑤 ∈ ω ∧ 𝑣N) → ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅))
2610, 5syl6eqel 2128 . . . . . . . 8 (𝑤 ∈ ω → (∅ ·𝑜 𝑤) ∈ ω)
27 enq0eceq 6535 . . . . . . . . 9 ((((∅ ·𝑜 𝑤) ∈ ω ∧ (1𝑜 ·𝑜 𝑣) ∈ N) ∧ (∅ ∈ ω ∧ 1𝑜N)) → ([⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 ↔ ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅)))
285, 6, 27mpanr12 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 ∅)))
2926, 20, 28syl2an 273 . . . . . . 7 ((𝑤 ∈ ω ∧ 𝑣N) → ([⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 ↔ ((∅ ·𝑜 𝑤) ·𝑜 1𝑜) = ((1𝑜 ·𝑜 𝑣) ·𝑜 ∅)))
3025, 29mpbird 156 . . . . . 6 ((𝑤 ∈ ω ∧ 𝑣N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = [⟨∅, 1𝑜⟩] ~Q0 )
3130, 2syl6eqr 2090 . . . . 5 ((𝑤 ∈ ω ∧ 𝑣N) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
3231adantr 261 . . . 4 (((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → [⟨(∅ ·𝑜 𝑤), (1𝑜 ·𝑜 𝑣)⟩] ~Q0 = 0Q0)
339, 32eqtrd 2072 . . 3 (((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
3433exlimivv 1776 . 2 (∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ) → (0Q0 ·Q0 𝐴) = 0Q0)
351, 34syl 14 1 (𝐴Q0 → (0Q0 ·Q0 𝐴) = 0Q0)
 Colors of variables: wff set class 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  1𝑜c1o 5994   ·𝑜 comu 5999  [cec 6104  Ncnpi 6370   ·N cmi 6372   ~Q0 ceq0 6384  Q0cnq0 6385  0Q0c0q0 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
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