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Theorem nqnq0m 7417
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 7340 . . . 4 (𝐴Q → ∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
2 nqpi 7340 . . . 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 1927 . . 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 5862 . . . . . . 7 ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q ) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ))
7 mulpiord 7279 . . . . . . . . . . 11 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
87ad2ant2r 506 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
9 mulpiord 7279 . . . . . . . . . . 11 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
109ad2ant2l 505 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
118, 10opeq12d 3773 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩ = ⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩)
1211eceq1d 6549 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
13 mulpipqqs 7335 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
14 mulclpi 7290 . . . . . . . . . . 11 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) ∈ N)
1514ad2ant2r 506 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑧 ·N 𝑣) ∈ N)
16 mulclpi 7290 . . . . . . . . . . 11 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
1716ad2ant2l 505 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝑤 ·N 𝑢) ∈ N)
18 nqnq0pi 7400 . . . . . . . . . 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 2206 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 )
21 pinn 7271 . . . . . . . . . 10 (𝑧N𝑧 ∈ ω)
2221anim1i 338 . . . . . . . . 9 ((𝑧N𝑤N) → (𝑧 ∈ ω ∧ 𝑤N))
23 pinn 7271 . . . . . . . . . 10 (𝑣N𝑣 ∈ ω)
2423anim1i 338 . . . . . . . . 9 ((𝑣N𝑢N) → (𝑣 ∈ ω ∧ 𝑢N))
25 mulnnnq0 7412 . . . . . . . . 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 2213 . . . . . . 7 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
286, 27sylan9eqr 2225 . . . . . 6 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
29 nqnq0pi 7400 . . . . . . . . . . 11 ((𝑧N𝑤N) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
3029adantr 274 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
3130eqeq2d 2182 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
32 nqnq0pi 7400 . . . . . . . . . . 11 ((𝑣N𝑢N) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
3332adantl 275 . . . . . . . . . 10 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
3433eqeq2d 2182 . . . . . . . . 9 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → (𝐵 = [⟨𝑣, 𝑢⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
3531, 34anbi12d 470 . . . . . . . 8 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
3635pm5.32i 451 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
37 oveq12 5862 . . . . . . . 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 2206 . . . . 5 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4140an4s 583 . . . 4 ((((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4241exlimivv 1889 . . 3 (∃𝑣𝑢(((𝑧N𝑤N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣N𝑢N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
4342exlimivv 1889 . 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 1348  wex 1485  wcel 2141  cop 3586  ωcom 4574  (class class class)co 5853   ·o comu 6393  [cec 6511  Ncnpi 7234   ·N cmi 7236   ~Q ceq 7241  Qcnq 7242   ·Q cmq 7245   ~Q0 ceq0 7248   ·Q0 cmq0 7252
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-enq0 7386  df-nq0 7387  df-mq0 7390
This theorem is referenced by:  prarloclemlo  7456  prarloclemcalc  7464
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