Theorem mulnqpr 7938

Description: Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqpr	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))

Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem mulnqpr

Step	Hyp	Ref	Expression
1		mulnqprlemfl 7936	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
2		mulnqprlemrl 7934	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
3	1, 2	eqssd 3265	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
4		mulnqprlemfu 7937	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
5		mulnqprlemru 7935	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
6	4, 5	eqssd 3265	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
7		mulclnq 7737	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
8		nqprlu 7908	. . . 4 ⊢ ((𝐴 ·Q 𝐵) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ ∈ P)
9	7, 8	syl 14	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ ∈ P)
10		nqprlu 7908	. . . 4 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
11		nqprlu 7908	. . . 4 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
12		mulclpr 7933	. . . 4 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P)
13	10, 11, 12	syl2an 289	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P)
14		preqlu 7833	. . 3 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ ∈ P ∧ (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ ((1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ∧ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))))
15	9, 13, 14	syl2anc 415	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ ((1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ∧ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))))
16	3, 6, 15	mpbir2and 957	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   ·_Q cmq 7644   <_Q cltq 7646  Pcnp 7652   ·_P cmp 7655

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-imp 7830

This theorem is referenced by:  recidpipr  8217