Theorem zeroopropd 50080

Description: Two structures with the same base, hom-sets and composition operation have the same zero objects. (Contributed by Zhi Wang, 26-Oct-2025.)

Hypotheses

Ref	Expression
initopropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
initopropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

Assertion

Ref	Expression
zeroopropd	⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

Proof of Theorem zeroopropd

Step	Hyp	Ref	Expression
1		initopropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
3		initopropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
5		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
6	2, 4, 5	zeroopropdlem 50077	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
7	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2771	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2771	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	zeroopropdlem 50077	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐷) = (ZeroO‘𝐶))
13	12	eqcomd 2771	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
14	1	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	3	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (compf‘𝐶) = (compf‘𝐷))
16	14, 15	initopropd 50078	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
17	14, 15	termopropd 50079	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
18	16, 17	ineq12d 4174	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((InitO‘𝐶) ∩ (TermO‘𝐶)) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
19		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
20		eqid 2765	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22	19, 20, 21	zerooval 18074	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
23	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (Homf ‘𝐶) = (Homf ‘𝐷))
24	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
25		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
26		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
27	23, 24, 25, 26	catpropd 17787	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
28	27	biimpa 482	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
29		eqid 2765	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
30		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	28, 29, 30	zerooval 18074	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐷) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
32	18, 22, 31	3eqtr4d 2810	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
33	27	pm5.32i 585	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
34	33, 32	sylbir 238	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
35		zeroofn 18068	. . . . . . . 8 ⊢ ZeroO Fn Cat
36	35	fndmi 6643	. . . . . . 7 ⊢ dom ZeroO = Cat
37	36	eleq2i 2857	. . . . . 6 ⊢ (𝐶 ∈ dom ZeroO ↔ 𝐶 ∈ Cat)
38		ndmfv 6917	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom ZeroO → (ZeroO‘𝐶) = ∅)
39	37, 38	sylnbir 334	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (ZeroO‘𝐶) = ∅)
40	39	ad2antrl 741	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐶) = ∅)
41	36	eleq2i 2857	. . . . . 6 ⊢ (𝐷 ∈ dom ZeroO ↔ 𝐷 ∈ Cat)
42		ndmfv 6917	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom ZeroO → (ZeroO‘𝐷) = ∅)
43	41, 42	sylnbir 334	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (ZeroO‘𝐷) = ∅)
44	43	ad2antll 742	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐷) = ∅)
45	40, 44	eqtr4d 2803	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
46	32, 34, 45	pm2.61ddan 826	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
47	6, 13, 46	pm2.61dda 827	1 ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905  ∅c0 4286  dom cdm 5663  ‘cfv 6540  Basecbs 17291  Hom chom 17343  Catccat 17742  Hom_f chomf 17744  comp_fccomf 17745  InitOcinito 18060  TermOctermo 18061  ZeroOczeroo 18062

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-cat 17746  df-homf 17748  df-comf 17749  df-inito 18063  df-termo 18064  df-zeroo 18065

This theorem is used by: (None)