Theorem mulcanenq0ec 7655

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 7645	. . 3 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ~Q0 Er (ω × N))
3		pinn 7519	. . . . 5 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4	3	3ad2ant1 1042	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐴 ∈ ω)
5		simp2 1022	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐵 ∈ ω)
6		pinn 7519	. . . . 5 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
7	6	3ad2ant3 1044	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐶 ∈ ω)
8		nnmcom 6652	. . . . 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 6650	. . . . 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 6198	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵))
13		nnmcl 6644	. . . . . . . 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 7527	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
16		mulclpi 7538	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
17	15, 16	eqeltrrd 2307	. . . . . . 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 590	. . . . . 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 1327	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
23		enq0breq 7646	. . . 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 6745	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 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3670   class class class wbr 4086  ωcom 4686   × cxp 4721  (class class class)co 6013   ·_o comu 6575   Er wer 6694  [cec 6695  Ncnpi 7482   ·_N cmi 7484   ~_Q0 ceq0 7496

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-ni 7514  df-mi 7516  df-enq0 7634

This theorem is referenced by:  nnanq0  7668  distrnq0  7669