Theorem mulcanenq0ec 7386

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 7376	. . 3 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ~Q0 Er (ω × N))
3		pinn 7250	. . . . 5 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4	3	3ad2ant1 1008	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐴 ∈ ω)
5		simp2 988	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐵 ∈ ω)
6		pinn 7250	. . . . 5 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
7	6	3ad2ant3 1010	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → 𝐶 ∈ ω)
8		nnmcom 6457	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
9	8	adantl 275	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
10		nnmass 6455	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧)))
11	10	adantl 275	. . . 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 6022	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) ·o 𝐵))
13		nnmcl 6449	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
14	3, 13	sylan 281	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
15		mulpiord 7258	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
16		mulclpi 7269	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
17	15, 16	eqeltrrd 2244	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·o 𝐶) ∈ N)
18	14, 17	anim12i 336	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N))
19		simpr 109	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐴 ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)) → (𝐵 ∈ ω ∧ 𝐶 ∈ N))
20	19	an4s 578	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → (𝐵 ∈ ω ∧ 𝐶 ∈ N))
21	18, 20	jca 304	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
22	21	3impdi 1283	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶 ∈ N)))
23		enq0breq 7377	. . . 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 166	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → ⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩ ~Q0 ⟨𝐵, 𝐶⟩)
26	2, 25	erthi 6547	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → [⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ⟨cop 3579   class class class wbr 3982  ωcom 4567   × cxp 4602  (class class class)co 5842   ·_o comu 6382   Er wer 6498  [cec 6499  Ncnpi 7213   ·_N cmi 7215   ~_Q0 ceq0 7227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-ni 7245  df-mi 7247  df-enq0 7365

This theorem is referenced by:  nnanq0  7399  distrnq0  7400