Theorem mulcanenq 11029

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 7456	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
2	1	opeq1d 4903	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3		opeq1 4897	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
4	2, 3	breq12d 5179	. . . . 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 7456	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
7	6	opeq2d 4904	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8		opeq2 4898	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
9	7, 8	breq12d 5179	. . . . 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 10965	. . . . . . . . 9 ⊢ (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
12	11	oveq2i 7459	. . . . . . . 8 ⊢ (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13		mulasspi 10966	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14		mulasspi 10966	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
15	12, 13, 14	3eqtr4i 2778	. . . . . . 7 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16		mulclpi 10962	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
17	16	3adant3 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
18		mulclpi 10962	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
19	18	3adant2 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
20		3simpc 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝑏 ∈ N ∧ 𝑐 ∈ N))
21		enqbreq 10988	. . . . . . . 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 837	. . . . . . 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 1120	. . . . 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 3590	. . 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 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  (class class class)co 7448  Ncnpi 10913   ·_N cmi 10915   ~_Q ceq 10920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526  df-omul 8527  df-ni 10941  df-mi 10943  df-enq 10980

This theorem is referenced by:  distrnq  11030  1nqenq  11031  ltexnq  11044