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

Theorem mulcanenq 10374
 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 7144 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
21opeq1d 4772 . . . . . 6 (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3 opeq1 4764 . . . . . 6 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
42, 3breq12d 5044 . . . . 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 7144 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
76opeq2d 4773 . . . . . 6 (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8 opeq2 4766 . . . . . 6 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
97, 8breq12d 5044 . . . . 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 10310 . . . . . . . . 9 (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
1211oveq2i 7147 . . . . . . . 8 (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13 mulasspi 10311 . . . . . . . 8 ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14 mulasspi 10311 . . . . . . . 8 ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
1512, 13, 143eqtr4i 2831 . . . . . . 7 ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16 mulclpi 10307 . . . . . . . . 9 ((𝐴N𝑏N) → (𝐴 ·N 𝑏) ∈ N)
17163adant3 1129 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑏) ∈ N)
18 mulclpi 10307 . . . . . . . . 9 ((𝐴N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
19183adant2 1128 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
20 3simpc 1147 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝑏N𝑐N))
21 enqbreq 10333 . . . . . . . 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 836 . . . . . . 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 1117 . . . . 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 3523 . . 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 1112 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 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5031  (class class class)co 7136  Ncnpi 10258   ·N cmi 10260   ~Q ceq 10265 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-oadd 8092  df-omul 8093  df-ni 10286  df-mi 10288  df-enq 10325 This theorem is referenced by:  distrnq  10375  1nqenq  10376  ltexnq  10389
 Copyright terms: Public domain W3C validator