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Theorem mulcanenq 11000
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 7439 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
21opeq1d 4879 . . . . . 6 (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3 opeq1 4873 . . . . . 6 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
42, 3breq12d 5156 . . . . 5 (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩))
54imbi2d 340 . . . 4 (𝑏 = 𝐵 → ((𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩)))
6 oveq2 7439 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
76opeq2d 4880 . . . . . 6 (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8 opeq2 4874 . . . . . 6 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
97, 8breq12d 5156 . . . . 5 (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
109imbi2d 340 . . . 4 (𝑐 = 𝐶 → ((𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)))
11 mulcompi 10936 . . . . . . . . 9 (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
1211oveq2i 7442 . . . . . . . 8 (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13 mulasspi 10937 . . . . . . . 8 ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14 mulasspi 10937 . . . . . . . 8 ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
1512, 13, 143eqtr4i 2775 . . . . . . 7 ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16 mulclpi 10933 . . . . . . . . 9 ((𝐴N𝑏N) → (𝐴 ·N 𝑏) ∈ N)
17163adant3 1133 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑏) ∈ N)
18 mulclpi 10933 . . . . . . . . 9 ((𝐴N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
19183adant2 1132 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
20 3simpc 1151 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝑏N𝑐N))
21 enqbreq 10959 . . . . . . . 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 838 . . . . . . 7 ((𝐴N𝑏N𝑐N) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
2315, 22mpbiri 258 . . . . . 6 ((𝐴N𝑏N𝑐N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
24233expb 1121 . . . . 5 ((𝐴N ∧ (𝑏N𝑐N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
2524expcom 413 . . . 4 ((𝑏N𝑐N) → (𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩))
265, 10, 25vtocl2ga 3578 . . 3 ((𝐵N𝐶N) → (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
2726impcom 407 . 2 ((𝐴N ∧ (𝐵N𝐶N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
28273impb 1115 1 ((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  (class class class)co 7431  Ncnpi 10884   ·N cmi 10886   ~Q ceq 10891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511  df-ni 10912  df-mi 10914  df-enq 10951
This theorem is referenced by:  distrnq  11001  1nqenq  11002  ltexnq  11015
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