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Theorem mulcmpblnq0 6426
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 5464 . 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 5999 . . . . . . . 8 ((A 𝜔 𝐹 𝜔) → (A ·𝑜 𝐹) 𝜔)
3 mulpiord 6301 . . . . . . . . 9 ((B N 𝐺 N) → (B ·N 𝐺) = (B ·𝑜 𝐺))
4 mulclpi 6312 . . . . . . . . 9 ((B N 𝐺 N) → (B ·N 𝐺) N)
53, 4eqeltrrd 2112 . . . . . . . 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 522 . . . . . 6 (((A 𝜔 B N) (𝐹 𝜔 𝐺 N)) → ((A ·𝑜 𝐹) 𝜔 (B ·𝑜 𝐺) N))
8 nnmcl 5999 . . . . . . . 8 ((𝐶 𝜔 𝑅 𝜔) → (𝐶 ·𝑜 𝑅) 𝜔)
9 mulpiord 6301 . . . . . . . . 9 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) = (𝐷 ·𝑜 𝑆))
10 mulclpi 6312 . . . . . . . . 9 ((𝐷 N 𝑆 N) → (𝐷 ·N 𝑆) N)
119, 10eqeltrrd 2112 . . . . . . . 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 522 . . . . . 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 522 . . . 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 6418 . . . 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 485 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → A 𝜔)
19 simprll 489 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐹 𝜔)
20 simplrr 488 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐷 N)
21 pinn 6293 . . . . . 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 6007 . . . . . 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 6005 . . . . . 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 492 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝑆 N)
28 pinn 6293 . . . . . 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 5999 . . . . . 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 5627 . . . 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 486 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → B N)
34 pinn 6293 . . . . . 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 490 . . . . . 6 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐺 N)
37 pinn 6293 . . . . . 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 487 . . . . 5 ((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → 𝐶 𝜔)
40 simprrl 491 . . . . 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 5627 . . . 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 2051 . . 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 884   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  𝜔com 4256  (class class class)co 5455   ·𝑜 comu 5938  Ncnpi 6256   ·N cmi 6258   ~Q0 ceq0 6270
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290  df-enq0 6406
This theorem is referenced by:  mulnq0mo  6430
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