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Theorem mulcanenq 9639
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 6535 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
21opeq1d 4340 . . . . . 6 (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3 opeq1 4334 . . . . . 6 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
42, 3breq12d 4590 . . . . 5 (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩))
54imbi2d 328 . . . 4 (𝑏 = 𝐵 → ((𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩)))
6 oveq2 6535 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
76opeq2d 4341 . . . . . 6 (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8 opeq2 4335 . . . . . 6 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
97, 8breq12d 4590 . . . . 5 (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
109imbi2d 328 . . . 4 (𝑐 = 𝐶 → ((𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q𝐵, 𝑐⟩) ↔ (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)))
11 mulcompi 9575 . . . . . . . . 9 (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
1211oveq2i 6538 . . . . . . . 8 (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13 mulasspi 9576 . . . . . . . 8 ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14 mulasspi 9576 . . . . . . . 8 ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
1512, 13, 143eqtr4i 2641 . . . . . . 7 ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16 mulclpi 9572 . . . . . . . . 9 ((𝐴N𝑏N) → (𝐴 ·N 𝑏) ∈ N)
17163adant3 1073 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑏) ∈ N)
18 mulclpi 9572 . . . . . . . . 9 ((𝐴N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
19183adant2 1072 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝐴 ·N 𝑐) ∈ N)
20 3simpc 1052 . . . . . . . 8 ((𝐴N𝑏N𝑐N) → (𝑏N𝑐N))
21 enqbreq 9598 . . . . . . . 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 1316 . . . . . . 7 ((𝐴N𝑏N𝑐N) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
2315, 22mpbiri 246 . . . . . 6 ((𝐴N𝑏N𝑐N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
24233expb 1257 . . . . 5 ((𝐴N ∧ (𝑏N𝑐N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩)
2524expcom 449 . . . 4 ((𝑏N𝑐N) → (𝐴N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q𝑏, 𝑐⟩))
265, 10, 25vtocl2ga 3246 . . 3 ((𝐵N𝐶N) → (𝐴N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩))
2726impcom 444 . 2 ((𝐴N ∧ (𝐵N𝐶N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
28273impb 1251 1 ((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577  (class class class)co 6527  Ncnpi 9523   ·N cmi 9525   ~Q ceq 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-oadd 7429  df-omul 7430  df-ni 9551  df-mi 9553  df-enq 9590
This theorem is referenced by:  distrnq  9640  1nqenq  9641  ltexnq  9654
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