Theorem mulcanenq0ec 7802

Description: Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)

Assertion

Ref	Expression
mulcanenq0ec	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → [⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Proof of Theorem mulcanenq0ec

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enq0er 7792	. . 3 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ~Q0 Er (ω × N))
3		pinn 7666	. . . . 5 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4	3	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐴 ∈ ω)
5		simp2 1029	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐵 ∈ ω)
6		pinn 7666	. . . . 5 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
7	6	3ad2ant3 1051	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐶 ∈ ω)
8		nnmcom 6752	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
9	8	adantl 277	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
10		nnmass 6750	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧)))
11	10	adantl 277	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧)))
12	4, 5, 7, 9, 11	caov32d 6260	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵))
13		nnmcl 6744	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
14	3, 13	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
15		mulpiord 7674	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
16		mulclpi 7685	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
17	15, 16	eqeltrrd 2316	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·o 𝐶) ∈ N)
18	14, 17	anim12i 338	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N))
19		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐴 ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)) → (𝐵 ∈ ω ∧ 𝐶 ∈ N))
20	19	an4s 596	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → (𝐵 ∈ ω ∧ 𝐶 ∈ N))
21	18, 20	jca 306	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
22	21	3impdi 1334	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
23		enq0breq 7793	. . . 4 ⊢ ((((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)) → (⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩ ~Q0 ⟨𝐵, 𝐶⟩ ↔ ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵)))
24	22, 23	syl 14	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → (⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩ ~Q0 ⟨𝐵, 𝐶⟩ ↔ ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵)))
25	12, 24	mpbird 167	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩ ~Q0 ⟨𝐵, 𝐶⟩)
26	2, 25	erthi 6845	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → [⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3708   class class class wbr 4125  ωcom 4732   × cxp 4767  (class class class)co 6075   ·_o comu 6675   Er wer 6794  [cec 6795  Ncnpi 7629   ·_N cmi 7631   ~_Q0 ceq0 7643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-ni 7661  df-mi 7663  df-enq0 7781

This theorem is referenced by:  nnanq0  7815  distrnq0  7816