Theorem iszeroi 17971

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 22	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
2		eqid 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	zerooval 17957	. . . . 5 ⊢ (𝐶 ∈ Cat → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5	4	eleq2d 2814	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) ↔ 𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6		elin 3930	. . . . 5 ⊢ (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)))
7		initoo 17969	. . . . . 6 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
8	7	adantrd 491	. . . . 5 ⊢ (𝐶 ∈ Cat → ((𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
9	6, 8	biimtrid 242	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
10	5, 9	sylbid 240	. . 3 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
11	10	imp 406	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
12		simpl 482	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13		simpr 484	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
14	2, 3, 12, 13	iszeroo 17960	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
15	14	biimpd 229	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
16	15	impancom 451	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
17	11, 16	jcai 516	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ∩ cin 3913  ‘cfv 6511  Basecbs 17179  Hom chom 17231  Catccat 17625  InitOcinito 17943  TermOctermo 17944  ZeroOczeroo 17945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-inito 17946  df-zeroo 17948

This theorem is referenced by:  nzerooringczr  21390  zeroo2  49223