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Theorem mulcmpblnq0 6599
Description: Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
Assertion
Ref Expression
mulcmpblnq0 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩))

Proof of Theorem mulcmpblnq0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 5548 . 2 (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑅)))
2 nnmcl 6090 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐹 ∈ ω) → (𝐴 ·𝑜 𝐹) ∈ ω)
3 mulpiord 6472 . . . . . . . . 9 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) = (𝐵 ·𝑜 𝐺))
4 mulclpi 6483 . . . . . . . . 9 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) ∈ N)
53, 4eqeltrrd 2131 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·𝑜 𝐺) ∈ N)
62, 5anim12i 325 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐹 ∈ ω) ∧ (𝐵N𝐺N)) → ((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N))
76an4s 530 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐹 ∈ ω ∧ 𝐺N)) → ((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N))
8 nnmcl 6090 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝑅 ∈ ω) → (𝐶 ·𝑜 𝑅) ∈ ω)
9 mulpiord 6472 . . . . . . . . 9 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
10 mulclpi 6483 . . . . . . . . 9 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) ∈ N)
119, 10eqeltrrd 2131 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·𝑜 𝑆) ∈ N)
128, 11anim12i 325 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝑅 ∈ ω) ∧ (𝐷N𝑆N)) → ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N))
1312an4s 530 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐷N) ∧ (𝑅 ∈ ω ∧ 𝑆N)) → ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N))
147, 13anim12i 325 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐹 ∈ ω ∧ 𝐺N)) ∧ ((𝐶 ∈ ω ∧ 𝐷N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N) ∧ ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N)))
1514an4s 530 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N) ∧ ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N)))
16 enq0breq 6591 . . . 4 ((((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N) ∧ ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N)) → (⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩ ↔ ((𝐴 ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑅))))
1715, 16syl 14 . . 3 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩ ↔ ((𝐴 ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑅))))
18 simplll 493 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐴 ∈ ω)
19 simprll 497 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐹 ∈ ω)
20 simplrr 496 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷N)
21 pinn 6464 . . . . . 6 (𝐷N𝐷 ∈ ω)
2220, 21syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷 ∈ ω)
23 nnmcom 6098 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
2423adantl 266 . . . . 5 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
25 nnmass 6096 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
2625adantl 266 . . . . 5 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
27 simprrr 500 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆N)
28 pinn 6464 . . . . . 6 (𝑆N𝑆 ∈ ω)
2927, 28syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆 ∈ ω)
30 nnmcl 6090 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) ∈ ω)
3130adantl 266 . . . . 5 (((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·𝑜 𝑦) ∈ ω)
3218, 19, 22, 24, 26, 29, 31caov4d 5712 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐴 ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)))
33 simpllr 494 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵N)
34 pinn 6464 . . . . . 6 (𝐵N𝐵 ∈ ω)
3533, 34syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵 ∈ ω)
36 simprlr 498 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺N)
37 pinn 6464 . . . . . 6 (𝐺N𝐺 ∈ ω)
3836, 37syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺 ∈ ω)
39 simplrl 495 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐶 ∈ ω)
40 simprrl 499 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑅 ∈ ω)
4135, 38, 39, 24, 26, 40, 31caov4d 5712 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑅)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑅)))
4232, 41eqeq12d 2070 . . 3 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑅)) ↔ ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑅))))
4317, 42bitrd 181 . 2 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩ ↔ ((𝐴 ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)) = ((𝐵 ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑅))))
441, 43syl5ibr 149 1 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  cop 3405   class class class wbr 3791  ωcom 4340  (class class class)co 5539   ·𝑜 comu 6029  Ncnpi 6427   ·N cmi 6429   ~Q0 ceq0 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-ni 6459  df-mi 6461  df-enq0 6579
This theorem is referenced by:  mulnq0mo  6603
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