Theorem mulcanenq 10883

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 7375	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 ·N 𝑏) = (𝐴 ·N 𝐵))
2	1	opeq1d 4822	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨(𝐴 ·N 𝑏), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩)
3		opeq1 4816	. . . . . 6 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
4	2, 3	breq12d 5098	. . . . 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 7375	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 ·N 𝑐) = (𝐴 ·N 𝐶))
7	6	opeq2d 4823	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝑐)⟩ = ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩)
8		opeq2 4817	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
9	7, 8	breq12d 5098	. . . . 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 10819	. . . . . . . . 9 ⊢ (𝑏 ·N 𝑐) = (𝑐 ·N 𝑏)
12	11	oveq2i 7378	. . . . . . . 8 ⊢ (𝐴 ·N (𝑏 ·N 𝑐)) = (𝐴 ·N (𝑐 ·N 𝑏))
13		mulasspi 10820	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = (𝐴 ·N (𝑏 ·N 𝑐))
14		mulasspi 10820	. . . . . . . 8 ⊢ ((𝐴 ·N 𝑐) ·N 𝑏) = (𝐴 ·N (𝑐 ·N 𝑏))
15	12, 13, 14	3eqtr4i 2769	. . . . . . 7 ⊢ ((𝐴 ·N 𝑏) ·N 𝑐) = ((𝐴 ·N 𝑐) ·N 𝑏)
16		mulclpi 10816	. . . . . . . . 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 10816	. . . . . . . . 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 10842	. . . . . . . 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 3521	. . 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 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  (class class class)co 7367  Ncnpi 10767   ·_N cmi 10769   ~_Q ceq 10774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-omul 8410  df-ni 10795  df-mi 10797  df-enq 10834

This theorem is referenced by:  distrnq  10884  1nqenq  10885  ltexnq  10898