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Theorem mulcanenq 9661
 Description: Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulcanenq ((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)

Proof of Theorem mulcanenq
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
21opeq1d 4346 . . . . . 6 (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3 opeq1 4340 . . . . . 6 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
42, 3breq12d 4596 . . . . 5 (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩))
54imbi2d 329 . . . 4 (𝑏 = 𝐵 → ((𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩)))
6 oveq2 6557 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
76opeq2d 4347 . . . . . 6 (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8 opeq2 4341 . . . . . 6 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
97, 8breq12d 4596 . . . . 5 (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
109imbi2d 329 . . . 4 (𝑐 = 𝐶 → ((𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)))
11 mulcompi 9597 . . . . . . . . 9 (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
1211oveq2i 6560 . . . . . . . 8 (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13 mulasspi 9598 . . . . . . . 8 ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14 mulasspi 9598 . . . . . . . 8 ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
1512, 13, 143eqtr4i 2642 . . . . . . 7 ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16 mulclpi 9594 . . . . . . . . 9 ((𝐴N𝑏N) → (𝐴 ·N 𝑏) ∈ N)
17163adant3 1074 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑏) ∈ N)
18 mulclpi 9594 . . . . . . . . 9 ((𝐴N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
19183adant2 1073 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
20 3simpc 1053 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝑏N𝑐N))
21 enqbreq 9620 . . . . . . . 8 ((((𝐴 ·N 𝑏) ∈ N ∧ (𝐴 ·N 𝑐) ∈ N) ∧ (𝑏N𝑐N)) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
2217, 19, 20, 21syl21anc 1317 . . . . . . 7 ((𝐴N𝑏N𝑐N) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
2315, 22mpbiri 247 . . . . . 6 ((𝐴N𝑏N𝑐N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
24233expb 1258 . . . . 5 ((𝐴N ∧ (𝑏N𝑐N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
2524expcom 450 . . . 4 ((𝑏N𝑐N) → (𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩))
265, 10, 25vtocl2ga 3247 . . 3 ((𝐵N𝐶N) → (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
2726impcom 445 . 2 ((𝐴N ∧ (𝐵N𝐶N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
28273impb 1252 1 ((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  (class class class)co 6549  Ncnpi 9545   ·N cmi 9547   ~Q ceq 9552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-ni 9573  df-mi 9575  df-enq 9612 This theorem is referenced by:  distrnq  9662  1nqenq  9663  ltexnq  9676
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