Theorem zeroopropd 48968

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 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
3		initopropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
5		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
6	2, 4, 5	zeroopropdlem 48965	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
7	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2740	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2740	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	zeroopropdlem 48965	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐷) = (ZeroO‘𝐶))
13	12	eqcomd 2740	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
14	1	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	3	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (compf‘𝐶) = (compf‘𝐷))
16	14, 15	initopropd 48966	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
17	14, 15	termopropd 48967	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
18	16, 17	ineq12d 4194	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → ((InitO‘𝐶) ∩ (TermO‘𝐶)) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
19		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
20		eqid 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22	19, 20, 21	zerooval 17993	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
23	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (Homf ‘𝐶) = (Homf ‘𝐷))
24	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (compf‘𝐶) = (compf‘𝐷))
25		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐶 ∈ V)
26		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → 𝐷 ∈ V)
27	23, 24, 25, 26	catpropd 17706	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
28	27	biimpa 476	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
29		eqid 2734	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
30		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	28, 29, 30	zerooval 17993	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐷) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
32	18, 22, 31	3eqtr4d 2779	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
33	27	pm5.32i 574	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) ↔ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat))
34	33, 32	sylbir 235	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐷 ∈ Cat) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
35		zeroofn 17987	. . . . . . . 8 ⊢ ZeroO Fn Cat
36	35	fndmi 6638	. . . . . . 7 ⊢ dom ZeroO = Cat
37	36	eleq2i 2825	. . . . . 6 ⊢ (𝐶 ∈ dom ZeroO ↔ 𝐶 ∈ Cat)
38		ndmfv 6907	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom ZeroO → (ZeroO‘𝐶) = ∅)
39	37, 38	sylnbir 331	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (ZeroO‘𝐶) = ∅)
40	39	ad2antrl 728	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐶) = ∅)
41	36	eleq2i 2825	. . . . . 6 ⊢ (𝐷 ∈ dom ZeroO ↔ 𝐷 ∈ Cat)
42		ndmfv 6907	. . . . . 6 ⊢ (¬ 𝐷 ∈ dom ZeroO → (ZeroO‘𝐷) = ∅)
43	41, 42	sylnbir 331	. . . . 5 ⊢ (¬ 𝐷 ∈ Cat → (ZeroO‘𝐷) = ∅)
44	43	ad2antll 729	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐷) = ∅)
45	40, 44	eqtr4d 2772	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ (¬ 𝐶 ∈ Cat ∧ ¬ 𝐷 ∈ Cat)) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
46	32, 34, 45	pm2.61ddan 813	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
47	6, 13, 46	pm2.61dda 814	1 ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457   ∩ cin 3923  ∅c0 4306  dom cdm 5651  ‘cfv 6527  Basecbs 17213  Hom chom 17267  Catccat 17661  Hom_f chomf 17663  comp_fccomf 17664  InitOcinito 17979  TermOctermo 17980  ZeroOczeroo 17981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-cat 17665  df-homf 17667  df-comf 17668  df-inito 17982  df-termo 17983  df-zeroo 17984

This theorem is referenced by: (None)