Theorem iszeroi 16659

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

Assertion

Ref	Expression
iszeroi	|- ( ( C e. Cat /\ O e. (ZeroO ` C ) ) -> ( O e. ( Base ` C ) /\ ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) ) )

Proof of Theorem iszeroi

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 |- ( C e. Cat -> C e. Cat )
2		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
3		eqid 2622	. . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
4	1, 2, 3	zerooval 16649	. . . . 5 |- ( C e. Cat -> (ZeroO ` C ) = ( (InitO ` C ) i^i (TermO ` C ) ) )
5	4	eleq2d 2687	. . . 4 |- ( C e. Cat -> ( O e. (ZeroO ` C ) <-> O e. ( (InitO ` C ) i^i (TermO ` C ) ) ) )
6		elin 3796	. . . . 5 |- ( O e. ( (InitO ` C ) i^i (TermO ` C ) ) <-> ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) )
7		initoo 16657	. . . . . 6 |- ( C e. Cat -> ( O e. (InitO ` C ) -> O e. ( Base ` C ) ) )
8	7	adantrd 484	. . . . 5 |- ( C e. Cat -> ( ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) -> O e. ( Base ` C ) ) )
9	6, 8	syl5bi 232	. . . 4 |- ( C e. Cat -> ( O e. ( (InitO ` C ) i^i (TermO ` C ) ) -> O e. ( Base ` C ) ) )
10	5, 9	sylbid 230	. . 3 |- ( C e. Cat -> ( O e. (ZeroO ` C ) -> O e. ( Base ` C ) ) )
11	10	imp 445	. 2 |- ( ( C e. Cat /\ O e. (ZeroO ` C ) ) -> O e. ( Base ` C ) )
12		simpl 473	. . . . 5 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> C e. Cat )
13		simpr 477	. . . . 5 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> O e. ( Base ` C ) )
14	2, 3, 12, 13	iszeroo 16652	. . . 4 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( O e. (ZeroO ` C ) <-> ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) ) )
15	14	biimpd 219	. . 3 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( O e. (ZeroO ` C ) -> ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) ) )
16	15	impancom 456	. 2 |- ( ( C e. Cat /\ O e. (ZeroO ` C ) ) -> ( O e. ( Base ` C ) -> ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) ) )
17	11, 16	jcai 559	1 |- ( ( C e. Cat /\ O e. (ZeroO ` C ) ) -> ( O e. ( Base ` C ) /\ ( O e. (InitO ` C ) /\ O e. (TermO ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   i^i cin 3573   ` cfv 5888   Basecbs 15857   Hom chom 15952   Catccat 16325  InitOcinito 16638  TermOctermo 16639  ZeroOczeroo 16640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-inito 16641  df-zeroo 16643

This theorem is referenced by:  nzerooringczr  42072