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	⊢ ((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 7439	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
2	1	opeq1d 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⟩)
4	2, 3	breq12d 5156	. . . . 5 ⊢ (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩))
5	4	imbi2d 340	. . . 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 7439	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
7	6	opeq2d 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⟩)
9	7, 8	breq12d 5156	. . . . 5 ⊢ (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
10	9	imbi2d 340	. . . 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 10936	. . . . . . . . 9 ⊢ (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
12	11	oveq2i 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))
15	12, 13, 14	3eqtr4i 2775	. . . . . . 7 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16		mulclpi 10933	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
17	16	3adant3 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
18		mulclpi 10933	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
19	18	3adant2 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
20		3simpc 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝑏 ∈ N ∧ 𝑐 ∈ N))
21		enqbreq 10959	. . . . . . . 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 838	. . . . . . 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 258	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
24	23	3expb 1121	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝑏 ∈ N ∧ 𝑐 ∈ N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
25	24	expcom 413	. . . 4 ⊢ ((𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩))
26	5, 10, 25	vtocl2ga 3578	. . 3 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
27	26	impcom 407	. 2 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)
28	27	3impb 1115	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)

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