Theorem iszeroi 17918

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 2733	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	zerooval 17904	. . . . 5 ⊢ (𝐶 ∈ Cat → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5	4	eleq2d 2819	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) ↔ 𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6		elin 3914	. . . . 5 ⊢ (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)))
7		initoo 17916	. . . . . 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 17907	. . . 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 2113   ∩ cin 3897  ‘cfv 6486  Basecbs 17122  Hom chom 17174  Catccat 17572  InitOcinito 17890  TermOctermo 17891  ZeroOczeroo 17892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-inito 17893  df-zeroo 17895

This theorem is referenced by:  nzerooringczr  21419  zeroo2  49359