Theorem mulcanenq0ec 7628

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 7618	. . 3 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ~Q0 Er (ω × N))
3		pinn 7492	. . . . 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 7492	. . . . 5 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
7	6	3ad2ant3 1044	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐶 ∈ ω)
8		nnmcom 6633	. . . . 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 6631	. . . . 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 6185	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵))
13		nnmcl 6625	. . . . . . . 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 7500	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
16		mulclpi 7511	. . . . . . . 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 7619	. . . 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 6726	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 3669   class class class wbr 4082  ωcom 4681   × cxp 4716  (class class class)co 6000   ·_o comu 6558   Er wer 6675  [cec 6676  Ncnpi 7455   ·_N cmi 7457   ~_Q0 ceq0 7469

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-ni 7487  df-mi 7489  df-enq0 7607

This theorem is referenced by:  nnanq0  7641  distrnq0  7642