Theorem iszeroi 18056

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 2765	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2765	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	zerooval 18042	. . . . 5 ⊢ (𝐶 ∈ Cat → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
5	4	eleq2d 2851	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) ↔ 𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6		elin 3923	. . . . 5 ⊢ (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)))
7		initoo 18054	. . . . . 6 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
8	7	adantrd 496	. . . . 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 411	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
12		simpl 487	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13		simpr 489	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
14	2, 3, 12, 13	iszeroo 18045	. . . 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 456	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
17	11, 16	jcai 525	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145   ∩ cin 3906  ‘cfv 6525  Basecbs 17259  Hom chom 17311  Catccat 17710  InitOcinito 18028  TermOctermo 18029  ZeroOczeroo 18030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-inito 18031  df-zeroo 18033

This theorem is referenced by:  nzerooringczr  21590  zeroo2  49863