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Theorem mulcmpblnq0 6299
Description: Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
Assertion
Ref Expression
mulcmpblnq0 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((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 5445 . 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 5975 . . . . . . . 8 ((A 𝜔 𝐹 𝜔) → (A ·𝑜 𝐹) 𝜔)
3 mulpiord 6177 . . . . . . . . 9 ((B N 𝐺 N) → (B ·N 𝐺) = (B ·𝑜 𝐺))
4 mulclpi 6188 . . . . . . . . 9 ((B N 𝐺 N) → (B ·N 𝐺) N)
53, 4eqeltrrd 2097 . . . . . . . 8 ((B N 𝐺 N) → (B ·𝑜 𝐺) N)
62, 5anim12i 321 . . . . . . 7 (((A 𝜔 𝐹 𝜔) (B N 𝐺 N)) → ((A ·𝑜 𝐹) 𝜔 (B ·𝑜 𝐺) N))
76an4s 509 . . . . . 6 (((A 𝜔 B N) (𝐹 𝜔 𝐺 N)) → ((A ·𝑜 𝐹) 𝜔 (B ·𝑜 𝐺) N))
8 nnmcl 5975 . . . . . . . 8 ((𝐶 𝜔 𝑅 𝜔) → (𝐶 ·𝑜 𝑅) 𝜔)
9 mulpiord 6177 . . . . . . . . 9 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
10 mulclpi 6188 . . . . . . . . 9 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) N)
119, 10eqeltrrd 2097 . . . . . . . 8 ((𝐷 N 𝑆 N) → (𝐷 ·𝑜 𝑆) N)
128, 11anim12i 321 . . . . . . 7 (((𝐶 𝜔 𝑅 𝜔) (𝐷 N 𝑆 N)) → ((𝐶 ·𝑜 𝑅) 𝜔 (𝐷 ·𝑜 𝑆) N))
1312an4s 509 . . . . . 6 (((𝐶 𝜔 𝐷 N) (𝑅 𝜔 𝑆 N)) → ((𝐶 ·𝑜 𝑅) 𝜔 (𝐷 ·𝑜 𝑆) N))
147, 13anim12i 321 . . . . 5 ((((A 𝜔 B N) (𝐹 𝜔 𝐺 N)) ((𝐶 𝜔 𝐷 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐹) 𝜔 (B ·𝑜 𝐺) N) ((𝐶 ·𝑜 𝑅) 𝜔 (𝐷 ·𝑜 𝑆) N)))
1514an4s 509 . . . 4 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((A ·𝑜 𝐹) 𝜔 (B ·𝑜 𝐺) N) ((𝐶 ·𝑜 𝑅) 𝜔 (𝐷 ·𝑜 𝑆) N)))
16 enq0breq 6291 . . . 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 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 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))))
18 simplll 473 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → A 𝜔)
19 simprll 477 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐹 𝜔)
20 simplrr 476 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 N)
21 pinn 6169 . . . . . 6 (𝐷 N𝐷 𝜔)
2220, 21syl 14 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 𝜔)
23 nnmcom 5983 . . . . . 6 ((x 𝜔 y 𝜔) → (x ·𝑜 y) = (y ·𝑜 x))
2423adantl 262 . . . . 5 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔)) → (x ·𝑜 y) = (y ·𝑜 x))
25 nnmass 5981 . . . . . 6 ((x 𝜔 y 𝜔 z 𝜔) → ((x ·𝑜 y) ·𝑜 z) = (x ·𝑜 (y ·𝑜 z)))
2625adantl 262 . . . . 5 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔 z 𝜔)) → ((x ·𝑜 y) ·𝑜 z) = (x ·𝑜 (y ·𝑜 z)))
27 simprrr 480 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 N)
28 pinn 6169 . . . . . 6 (𝑆 N𝑆 𝜔)
2927, 28syl 14 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 𝜔)
30 nnmcl 5975 . . . . . 6 ((x 𝜔 y 𝜔) → (x ·𝑜 y) 𝜔)
3130adantl 262 . . . . 5 (((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) (x 𝜔 y 𝜔)) → (x ·𝑜 y) 𝜔)
3218, 19, 22, 24, 26, 29, 31caov4d 5608 . . . 4 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((A ·𝑜 𝐹) ·𝑜 (𝐷 ·𝑜 𝑆)) = ((A ·𝑜 𝐷) ·𝑜 (𝐹 ·𝑜 𝑆)))
33 simpllr 474 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B N)
34 pinn 6169 . . . . . 6 (B NB 𝜔)
3533, 34syl 14 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B 𝜔)
36 simprlr 478 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 N)
37 pinn 6169 . . . . . 6 (𝐺 N𝐺 𝜔)
3836, 37syl 14 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 𝜔)
39 simplrl 475 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐶 𝜔)
40 simprrl 479 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑅 𝜔)
4135, 38, 39, 24, 26, 40, 31caov4d 5608 . . . 4 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → ((B ·𝑜 𝐺) ·𝑜 (𝐶 ·𝑜 𝑅)) = ((B ·𝑜 𝐶) ·𝑜 (𝐺 ·𝑜 𝑅)))
4232, 41eqeq12d 2036 . . 3 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((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 177 . 2 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (⟨(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 145 1 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((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 97  wb 98   w3a 873   = wceq 1228   wcel 1374  cop 3353   class class class wbr 3738  𝜔com 4240  (class class class)co 5436   ·𝑜 comu 5914  Ncnpi 6130   ·N cmi 6132   ~Q0 ceq0 6144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-ni 6164  df-mi 6166  df-enq0 6279
This theorem is referenced by:  mulnq0mo  6303
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