Theorem mulcanenq 11084

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	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)

Proof of Theorem mulcanenq

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7459	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
2	1	opeq1d 4851	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3		opeq1 4845	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
4	2, 3	breq12d 5121	. . . . 5 ⊢ (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩))
5	4	imbi2d 343	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩) ↔ (𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩)))
6		oveq2 7459	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
7	6	opeq2d 4852	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8		opeq2 4846	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
9	7, 8	breq12d 5121	. . . . 5 ⊢ (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
10	9	imbi2d 343	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩) ↔ (𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)))
11		mulcompi 11020	. . . . . . . . 9 ⊢ (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
12	11	oveq2i 7462	. . . . . . . 8 ⊢ (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13		mulasspi 11021	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14		mulasspi 11021	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
15	12, 13, 14	3eqtr4i 2796	. . . . . . 7 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16		mulclpi 11017	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
17	16	3adant3 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
18		mulclpi 11017	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
19	18	3adant2 1149	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
20		3simpc 1168	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝑏 ∈ N ∧ 𝑐 ∈ N))
21		enqbreq 11043	. . . . . . . 8 ⊢ ((((𝐴 ·N 𝑏) ∈ N ∧ (𝐴 ·N 𝑐) ∈ N) ∧ (𝑏 ∈ N ∧ 𝑐 ∈ N)) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
22	17, 19, 20, 21	syl21anc 851	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩ ↔ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)))
23	15, 22	mpbiri 261	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
24	23	3expb 1138	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝑏 ∈ N ∧ 𝑐 ∈ N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
25	24	expcom 419	. . . 4 ⊢ ((𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩))
26	5, 10, 25	vtocl2ga 3541	. . 3 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
27	26	impcom 413	. 2 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)
28	27	3impb 1132	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ⟨cop 4840   class class class wbr 5108  (class class class)co 7451  Ncnpi 10968   ·_N cmi 10970   ~_Q ceq 10975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-oadd 8518  df-omul 8519  df-ni 10996  df-mi 10998  df-enq 11035

This theorem is used by:  distrnq  11085  1nqenq  11086  ltexnq  11099