Theorem zeroopropd 49277

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 49274	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → (ZeroO‘𝐶) = (ZeroO‘𝐷))
7	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐶) = (Homf ‘𝐷))
8	7	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (Homf ‘𝐷) = (Homf ‘𝐶))
9	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐶) = (compf‘𝐷))
10	9	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (compf‘𝐷) = (compf‘𝐶))
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → ¬ 𝐷 ∈ V)
12	8, 10, 11	zeroopropdlem 49274	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 ∈ V) → (ZeroO‘𝐷) = (ZeroO‘𝐶))
13	12	eqcomd 2737	. 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 49275	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (InitO‘𝐶) = (InitO‘𝐷))
17	14, 15	termopropd 49276	. . . . 5 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (TermO‘𝐶) = (TermO‘𝐷))
18	16, 17	ineq12d 4166	. . . 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 2731	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		eqid 2731	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22	19, 20, 21	zerooval 17897	. . . 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 17610	. . . . . 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 2731	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
30		eqid 2731	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	28, 29, 30	zerooval 17897	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) ∧ 𝐶 ∈ Cat) → (ZeroO‘𝐷) = ((InitO‘𝐷) ∩ (TermO‘𝐷)))
32	18, 22, 31	3eqtr4d 2776	. . 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 17891	. . . . . . . 8 ⊢ ZeroO Fn Cat
36	35	fndmi 6580	. . . . . . 7 ⊢ dom ZeroO = Cat
37	36	eleq2i 2823	. . . . . 6 ⊢ (𝐶 ∈ dom ZeroO ↔ 𝐶 ∈ Cat)
38		ndmfv 6849	. . . . . 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 2823	. . . . . 6 ⊢ (𝐷 ∈ dom ZeroO ↔ 𝐷 ∈ Cat)
42		ndmfv 6849	. . . . . 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 2769	. . 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 2111  Vcvv 3436   ∩ cin 3896  ∅c0 4278  dom cdm 5611  ‘cfv 6476  Basecbs 17115  Hom chom 17167  Catccat 17565  Hom_f chomf 17567  comp_fccomf 17568  InitOcinito 17883  TermOctermo 17884  ZeroOczeroo 17885

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-cat 17569  df-homf 17571  df-comf 17572  df-inito 17886  df-termo 17887  df-zeroo 17888

This theorem is referenced by: (None)