Theorem iszeroi 18189

Description: Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.)

Assertion

Ref	Expression
iszeroi	⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))

Proof of Theorem iszeroi

Step	Hyp	Ref	Expression
1		id 23	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
2		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	zerooval 18175	. . . . 5 ⊢ (𝐶 ∈ Cat → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5	4	eleq2d 2846	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) ↔ 𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6		elin 4093	. . . . 5 ⊢ (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)))
7		initoo 18187	. . . . . 6 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
8	7	adantrd 497	. . . . 5 ⊢ (𝐶 ∈ Cat → ((𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
9	6, 8	biimtrid 245	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
10	5, 9	sylbid 243	. . 3 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
11	10	imp 412	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
12		simpl 488	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13		simpr 490	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
14	2, 3, 12, 13	iszeroo 18178	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
15	14	biimpd 232	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
16	15	impancom 457	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
17	11, 16	jcai 526	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   ∩ cin 4089  ‘cfv 6538  Basecbs 17392  Hom chom 17444  Catccat 17843  InitOcinito 18161  TermOctermo 18162  ZeroOczeroo 18163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-inito 18164  df-zeroo 18166

This theorem is used by:  nzerooringczr  21791  zeroo2  50449