Theorem zeroopropd 49406

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 49403	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
7	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2739	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2739	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	zeroopropdlem 49403	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐷) = (ZeroO‘𝐶))
13	12	eqcomd 2739	. 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 49404	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
17	14, 15	termopropd 49405	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
18	16, 17	ineq12d 4170	. . . 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 2733	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		eqid 2733	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22	19, 20, 21	zerooval 17910	. . . 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 17623	. . . . . 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 2733	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
30		eqid 2733	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	28, 29, 30	zerooval 17910	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐷) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
32	18, 22, 31	3eqtr4d 2778	. . 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 17904	. . . . . . . 8 ⊢ ZeroO Fn Cat
36	35	fndmi 6593	. . . . . . 7 ⊢ dom ZeroO = Cat
37	36	eleq2i 2825	. . . . . 6 ⊢ (𝐶 ∈ dom ZeroO ↔ 𝐶 ∈ Cat)
38		ndmfv 6863	. . . . . 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 6863	. . . . . 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 2771	. . 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 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∅c0 4282  dom cdm 5621  ‘cfv 6489  Basecbs 17127  Hom chom 17179  Catccat 17578  Hom_f chomf 17580  comp_fccomf 17581  InitOcinito 17896  TermOctermo 17897  ZeroOczeroo 17898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-cat 17582  df-homf 17584  df-comf 17585  df-inito 17899  df-termo 17900  df-zeroo 17901

This theorem is referenced by: (None)