MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iszeroi Structured version   Visualization version   GIF version

Theorem iszeroi 16431
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
StepHypRef Expression
1 id 22 . . . . . 6 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
2 eqid 2610 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2610 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
41, 2, 3zerooval 16421 . . . . 5 (𝐶 ∈ Cat → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
54eleq2d 2673 . . . 4 (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) ↔ 𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶))))
6 elin 3758 . . . . 5 (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)))
7 initoo 16429 . . . . . 6 (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
87adantrd 483 . . . . 5 (𝐶 ∈ Cat → ((𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
96, 8syl5bi 231 . . . 4 (𝐶 ∈ Cat → (𝑂 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)))
105, 9sylbid 229 . . 3 (𝐶 ∈ Cat → (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
1110imp 444 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
12 simpl 472 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13 simpr 476 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
142, 3, 12, 13iszeroo 16424 . . . 4 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) ↔ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
1514biimpd 218 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → (𝑂 ∈ (ZeroO‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
1615impancom 455 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) → (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
1711, 16jcai 557 1 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  cin 3539  cfv 5790  Basecbs 15644  Hom chom 15728  Catccat 16097  InitOcinito 16410  TermOctermo 16411  ZeroOczeroo 16412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-inito 16413  df-zeroo 16415
This theorem is referenced by:  nzerooringczr  41856
  Copyright terms: Public domain W3C validator