Theorem mulcanenq 10874

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 7364	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
2	1	opeq1d 4810	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3		opeq1 4804	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
4	2, 3	breq12d 5085	. . . . 5 ⊢ (𝑏 = 𝐵 → (⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩))
5	4	imbi2d 341	. . . 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 7364	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
7	6	opeq2d 4811	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8		opeq2 4805	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
9	7, 8	breq12d 5085	. . . . 5 ⊢ (𝑐 = 𝐶 → (⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝐵, 𝑐⟩ ↔ ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
10	9	imbi2d 341	. . . 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 10810	. . . . . . . . 9 ⊢ (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
12	11	oveq2i 7367	. . . . . . . 8 ⊢ (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13		mulasspi 10811	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14		mulasspi 10811	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
15	12, 13, 14	3eqtr4i 2772	. . . . . . 7 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16		mulclpi 10807	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
17	16	3adant3 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑏) ∈ N)
18		mulclpi 10807	. . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
19	18	3adant2 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ·N 𝑐) ∈ N)
20		3simpc 1156	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝑏 ∈ N ∧ 𝑐 ∈ N))
21		enqbreq 10833	. . . . . . . 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 843	. . . . . . 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 259	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝑏 ∈ N ∧ 𝑐 ∈ N) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
24	23	3expb 1126	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝑏 ∈ N ∧ 𝑐 ∈ N)) → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩)
25	24	expcom 414	. . . 4 ⊢ ((𝑏 ∈ N ∧ 𝑐 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ ~Q ⟨𝑏, 𝑐⟩))
26	5, 10, 25	vtocl2ga 3521	. . 3 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩))
27	26	impcom 408	. 2 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)
28	27	3impb 1120	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   class class class wbr 5072  (class class class)co 7356  Ncnpi 10758   ·_N cmi 10760   ~_Q ceq 10765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399  df-omul 8400  df-ni 10786  df-mi 10788  df-enq 10825

This theorem is referenced by:  distrnq  10875  1nqenq  10876  ltexnq  10889