Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nqnq0m GIF version

Theorem nqnq0m 7307
 Description: Multiplication of positive fractions is equal with ·Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
Assertion
Ref Expression
nqnq0m ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))

Proof of Theorem nqnq0m
Dummy variables 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqpi 7230 . . . 4 (𝐴Q → ∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
2 nqpi 7230 . . . 4 (𝐵Q → ∃𝑣𝑢((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
31, 2anim12i 336 . . 3 ((𝐴Q𝐵Q) → (∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣𝑢((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
4 ee4anv 1907 . . 3 (∃𝑧𝑤𝑣𝑢(((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) ↔ (∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣𝑢((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
53, 4sylibr 133 . 2 ((𝐴Q𝐵Q) → ∃𝑧𝑤𝑣𝑢(((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
6 oveq12 5792 . . . . . . 7 ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q ) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ))
7 mulpiord 7169 . . . . . . . . . . 11 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
87ad2ant2r 501 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
9 mulpiord 7169 . . . . . . . . . . 11 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
109ad2ant2l 500 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
118, 10opeq12d 3722 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩ = ⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩)
1211eceq1d 6474 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
13 mulpipqqs 7225 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
14 mulclpi 7180 . . . . . . . . . . 11 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) ∈ N)
1514ad2ant2r 501 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑧 ·N 𝑣) ∈ N)
16 mulclpi 7180 . . . . . . . . . . 11 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
1716ad2ant2l 500 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑤 ·N 𝑢) ∈ N)
18 nqnq0pi 7290 . . . . . . . . . 10 (((𝑧 ·N 𝑣) ∈ N ∧ (𝑤 ·N 𝑢) ∈ N) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
1915, 17, 18syl2anc 409 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
2013, 19eqtr4d 2176 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 )
21 pinn 7161 . . . . . . . . . 10 (𝑧N𝑧 ∈ ω)
2221anim1i 338 . . . . . . . . 9 ((𝑧N𝑤N) → (𝑧 ∈ ω ∧ 𝑤N))
23 pinn 7161 . . . . . . . . . 10 (𝑣N𝑣 ∈ ω)
2423anim1i 338 . . . . . . . . 9 ((𝑣N𝑢N) → (𝑣 ∈ ω ∧ 𝑢N))
25 mulnnnq0 7302 . . . . . . . . 9 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
2622, 24, 25syl2an 287 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
2712, 20, 263eqtr4d 2183 . . . . . . 7 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
286, 27sylan9eqr 2195 . . . . . 6 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
29 nqnq0pi 7290 . . . . . . . . . . 11 ((𝑧N𝑤N) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
3029adantr 274 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
3130eqeq2d 2152 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
32 nqnq0pi 7290 . . . . . . . . . . 11 ((𝑣N𝑢N) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
3332adantl 275 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
3433eqeq2d 2152 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝐵 = [⟨𝑣, 𝑢⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
3531, 34anbi12d 465 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
3635pm5.32i 450 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
37 oveq12 5792 . . . . . . . 8 ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
3837adantl 275 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
3936, 38sylbir 134 . . . . . 6 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
4028, 39eqtr4d 2176 . . . . 5 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4140an4s 578 . . . 4 ((((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4241exlimivv 1869 . . 3 (∃𝑣𝑢(((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4342exlimivv 1869 . 2 (∃𝑧𝑤𝑣𝑢(((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
445, 43syl 14 1 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ⟨cop 3536  ωcom 4513  (class class class)co 5783   ·o comu 6320  [cec 6436  Ncnpi 7124   ·N cmi 7126   ~Q ceq 7131  Qcnq 7132   ·Q cmq 7135   ~Q0 ceq0 7138   ·Q0 cmq0 7142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-recs 6211  df-irdg 6276  df-oadd 6326  df-omul 6327  df-er 6438  df-ec 6440  df-qs 6444  df-ni 7156  df-mi 7158  df-mpq 7197  df-enq 7199  df-nqqs 7200  df-mqqs 7202  df-enq0 7276  df-nq0 7277  df-mq0 7280 This theorem is referenced by:  prarloclemlo  7346  prarloclemcalc  7354
 Copyright terms: Public domain W3C validator