MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulcanenq Structured version   Visualization version   GIF version

Theorem mulcanenq 10574
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 7221 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
21opeq1d 4790 . . . . . 6 (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3 opeq1 4784 . . . . . 6 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
42, 3breq12d 5066 . . . . 5 (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩))
54imbi2d 344 . . . 4 (𝑏 = 𝐵 → ((𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩)))
6 oveq2 7221 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
76opeq2d 4791 . . . . . 6 (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8 opeq2 4785 . . . . . 6 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
97, 8breq12d 5066 . . . . 5 (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
109imbi2d 344 . . . 4 (𝑐 = 𝐶 → ((𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)))
11 mulcompi 10510 . . . . . . . . 9 (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
1211oveq2i 7224 . . . . . . . 8 (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13 mulasspi 10511 . . . . . . . 8 ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14 mulasspi 10511 . . . . . . . 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 10507 . . . . . . . . 9 ((𝐴N𝑏N) → (𝐴 ·N 𝑏) ∈ N)
17163adant3 1134 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑏) ∈ N)
18 mulclpi 10507 . . . . . . . . 9 ((𝐴N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
19183adant2 1133 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
20 3simpc 1152 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝑏N𝑐N))
21 enqbreq 10533 . . . . . . . 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 261 . . . . . 6 ((𝐴N𝑏N𝑐N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
24233expb 1122 . . . . 5 ((𝐴N ∧ (𝑏N𝑐N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
2524expcom 417 . . . 4 ((𝑏N𝑐N) → (𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩))
265, 10, 25vtocl2ga 3490 . . 3 ((𝐵N𝐶N) → (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
2726impcom 411 . 2 ((𝐴N ∧ (𝐵N𝐶N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
28273impb 1117 1 ((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  cop 4547   class class class wbr 5053  (class class class)co 7213  Ncnpi 10458   ·N cmi 10460   ~Q ceq 10465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-oadd 8206  df-omul 8207  df-ni 10486  df-mi 10488  df-enq 10525
This theorem is referenced by:  distrnq  10575  1nqenq  10576  ltexnq  10589
  Copyright terms: Public domain W3C validator