Theorem nqnq0m 7680

Description: Multiplication of positive fractions is equal with ·_Q or ·_Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)

Assertion

Ref	Expression
nqnq0m	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (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.

Step	Hyp	Ref	Expression
1		nqpi 7603	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
2		nqpi 7603	. . . 4 ⊢ (𝐵 ∈ Q → ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
3	1, 2	anim12i 338	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
4		ee4anv 1986	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) ↔ (∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
5	3, 4	sylibr 134	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
6		oveq12 6032	. . . . . . 7 ⊢ ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ))
7		mulpiord 7542	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑣 ∈ N) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
8	7	ad2ant2r 509	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ·N 𝑣) = (𝑧 ·o 𝑣))
9		mulpiord 7542	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
10	9	ad2ant2l 508	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑢) = (𝑤 ·o 𝑢))
11	8, 10	opeq12d 3871	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩ = ⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩)
12	11	eceq1d 6743	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
13		mulpipqqs 7598	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
14		mulclpi 7553	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑣 ∈ N) → (𝑧 ·N 𝑣) ∈ N)
15	14	ad2ant2r 509	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ·N 𝑣) ∈ N)
16		mulclpi 7553	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·N 𝑢) ∈ N)
17	16	ad2ant2l 508	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑢) ∈ N)
18		nqnq0pi 7663	. . . . . . . . . 10 ⊢ (((𝑧 ·N 𝑣) ∈ N ∧ (𝑤 ·N 𝑢) ∈ N) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
19	15, 17, 18	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q )
20	13, 19	eqtr4d 2266	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨(𝑧 ·N 𝑣), (𝑤 ·N 𝑢)⟩] ~Q0 )
21		pinn 7534	. . . . . . . . . 10 ⊢ (𝑧 ∈ N → 𝑧 ∈ ω)
22	21	anim1i 340	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → (𝑧 ∈ ω ∧ 𝑤 ∈ N))
23		pinn 7534	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → 𝑣 ∈ ω)
24	23	anim1i 340	. . . . . . . . 9 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → (𝑣 ∈ ω ∧ 𝑢 ∈ N))
25		mulnnnq0 7675	. . . . . . . . 9 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
26	22, 24, 25	syl2an 289	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑧 ·o 𝑣), (𝑤 ·o 𝑢)⟩] ~Q0 )
27	12, 20, 26	3eqtr4d 2273	. . . . . . 7 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q ·Q [⟨𝑣, 𝑢⟩] ~Q ) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
28	6, 27	sylan9eqr 2285	. . . . . 6 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
29		nqnq0pi 7663	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
30	29	adantr 276	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
31	30	eqeq2d 2242	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ↔ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
32		nqnq0pi 7663	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
33	32	adantl 277	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
34	33	eqeq2d 2242	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ↔ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
35	31, 34	anbi12d 473	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
36	35	pm5.32i 454	. . . . . . 7 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
37		oveq12 6032	. . . . . . . 8 ⊢ ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
38	37	adantl 277	. . . . . . 7 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
39	36, 38	sylbir 135	. . . . . 6 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
40	28, 39	eqtr4d 2266	. . . . 5 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
41	40	an4s 592	. . . 4 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
42	41	exlimivv 1944	. . 3 ⊢ (∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
43	42	exlimivv 1944	. 2 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
44	5, 43	syl 14	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ⟨cop 3673  ωcom 4690  (class class class)co 6023   ·_o comu 6585  [cec 6705  Ncnpi 7497   ·_N cmi 7499   ~_Q ceq 7504  Qcnq 7505   ·_Q cmq 7508   ~_Q0 ceq0 7511   ·_Q0 cmq0 7515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-mi 7531  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-mqqs 7575  df-enq0 7649  df-nq0 7650  df-mq0 7653

This theorem is referenced by:  prarloclemlo  7719  prarloclemcalc  7727