Theorem mulcanenq0ec 7512

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 7502	. . 3 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ~Q0 Er (ω × N))
3		pinn 7376	. . . . 5 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4	3	3ad2ant1 1020	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐴 ∈ ω)
5		simp2 1000	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐵 ∈ ω)
6		pinn 7376	. . . . 5 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
7	6	3ad2ant3 1022	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐶 ∈ ω)
8		nnmcom 6547	. . . . 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 6545	. . . . 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 6104	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵))
13		nnmcl 6539	. . . . . . . 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 7384	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
16		mulclpi 7395	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
17	15, 16	eqeltrrd 2274	. . . . . . 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 588	. . . . . 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 1304	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
23		enq0breq 7503	. . . 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 6640	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 980   = wceq 1364   ∈ wcel 2167  ⟨cop 3625   class class class wbr 4033  ωcom 4626   × cxp 4661  (class class class)co 5922   ·_o comu 6472   Er wer 6589  [cec 6590  Ncnpi 7339   ·_N cmi 7341   ~_Q0 ceq0 7353

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-ni 7371  df-mi 7373  df-enq0 7491

This theorem is referenced by:  nnanq0  7525  distrnq0  7526