ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulcmpblnq0 GIF version

Theorem mulcmpblnq0 6982
Description: Lemma showing compatibility of multiplication on nonnegative 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 5643 . 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 6224 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐹 ∈ ω) → (𝐴 ·𝑜 𝐹) ∈ ω)
3 mulpiord 6855 . . . . . . . . 9 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) = (𝐵 ·𝑜 𝐺))
4 mulclpi 6866 . . . . . . . . 9 ((𝐵N𝐺N) → (𝐵 ·N 𝐺) ∈ N)
53, 4eqeltrrd 2165 . . . . . . . 8 ((𝐵N𝐺N) → (𝐵 ·𝑜 𝐺) ∈ N)
62, 5anim12i 331 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐹 ∈ ω) ∧ (𝐵N𝐺N)) → ((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N))
76an4s 555 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐹 ∈ ω ∧ 𝐺N)) → ((𝐴 ·𝑜 𝐹) ∈ ω ∧ (𝐵 ·𝑜 𝐺) ∈ N))
8 nnmcl 6224 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝑅 ∈ ω) → (𝐶 ·𝑜 𝑅) ∈ ω)
9 mulpiord 6855 . . . . . . . . 9 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
10 mulclpi 6866 . . . . . . . . 9 ((𝐷N𝑆N) → (𝐷 ·N 𝑆) ∈ N)
119, 10eqeltrrd 2165 . . . . . . . 8 ((𝐷N𝑆N) → (𝐷 ·𝑜 𝑆) ∈ N)
128, 11anim12i 331 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝑅 ∈ ω) ∧ (𝐷N𝑆N)) → ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N))
1312an4s 555 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐷N) ∧ (𝑅 ∈ ω ∧ 𝑆N)) → ((𝐶 ·𝑜 𝑅) ∈ ω ∧ (𝐷 ·𝑜 𝑆) ∈ N))
147, 13anim12i 331 . . . . 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 555 . . . 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 6974 . . . 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 500 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐴 ∈ ω)
19 simprll 504 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐹 ∈ ω)
20 simplrr 503 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷N)
21 pinn 6847 . . . . . 6 (𝐷N𝐷 ∈ ω)
2220, 21syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐷 ∈ ω)
23 nnmcom 6232 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
2423adantl 271 . . . . 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 6230 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
2625adantl 271 . . . . 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 507 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆N)
28 pinn 6847 . . . . . 6 (𝑆N𝑆 ∈ ω)
2927, 28syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑆 ∈ ω)
30 nnmcl 6224 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) ∈ ω)
3130adantl 271 . . . . 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 5811 . . . 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 501 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵N)
34 pinn 6847 . . . . . 6 (𝐵N𝐵 ∈ ω)
3533, 34syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐵 ∈ ω)
36 simprlr 505 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺N)
37 pinn 6847 . . . . . 6 (𝐺N𝐺 ∈ ω)
3836, 37syl 14 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐺 ∈ ω)
39 simplrl 502 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝐶 ∈ ω)
40 simprrl 506 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → 𝑅 ∈ ω)
4135, 38, 39, 24, 26, 40, 31caov4d 5811 . . . 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 2102 . . 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 186 . 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 154 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 102  wb 103  w3a 924   = wceq 1289  wcel 1438  cop 3444   class class class wbr 3837  ωcom 4395  (class class class)co 5634   ·𝑜 comu 6161  Ncnpi 6810   ·N cmi 6812   ~Q0 ceq0 6824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168  df-ni 6842  df-mi 6844  df-enq0 6962
This theorem is referenced by:  mulnq0mo  6986
  Copyright terms: Public domain W3C validator