Theorem zrtermoringc 20588

Description: The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)

Hypotheses

Ref	Expression
zrtermoringc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
zrtermoringc.c	⊢ 𝐶 = (RingCat‘𝑈)
zrtermoringc.z	⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
zrtermoringc.e	⊢ (𝜑 → 𝑍 ∈ 𝑈)

Assertion

Ref	Expression
zrtermoringc	⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Proof of Theorem zrtermoringc

Dummy variables 𝑎 ℎ 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zrtermoringc.c	. . . . . . . . . 10 ⊢ 𝐶 = (RingCat‘𝑈)
2		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrtermoringc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	ringcbas 20563	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
5	4	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6		elin 3918	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Ring))
7	6	simprbi 496	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
8	5, 7	biimtrdi 253	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
9	8	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10		zrtermoringc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑟) = (Base‘𝑟)
13		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑍) = (0g‘𝑍)
14		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))
15	12, 13, 14	c0rhm 20447	. . . . . 6 ⊢ ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
16	9, 11, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
18	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21		zrtermoringc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	10	eldifad 3914	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
23	21, 22	elind 4150	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
24	23, 4	eleqtrrd 2834	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
26	1, 2, 18, 19, 20, 25	ringchom 20565	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
27	26	eqcomd 2737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
28	27	eleq2d 2817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
29	28	biimpa 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
30	26	eleq2d 2817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ℎ ∈ (𝑟 RingHom 𝑍)))
31		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	12, 31	rhmf 20400	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑟 RingHom 𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍))
33	30, 32	biimtrdi 253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
34	33	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
35		ffn 6651	. . . . . . . . . . 11 ⊢ (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ Fn (Base‘𝑟))
36	35	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ Fn (Base‘𝑟))
37		fvex 6835	. . . . . . . . . . . 12 ⊢ (0g‘𝑍) ∈ V
38	37, 14	fnmpti 6624	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟)
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟))
40	31, 13	0ringbas 20441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
41	10, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
42	41	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g‘𝑍)})
43	42	feq3d 6636	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) ↔ ℎ:(Base‘𝑟)⟶{(0g‘𝑍)}))
44		fvconst 7096	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
45	44	ex 412	. . . . . . . . . . . . . 14 ⊢ (ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍)))
46	43, 45	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
47	46	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
48	47	imp31 417	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
49		eqidd 2732	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
50		eqidd 2732	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (Base‘𝑟) ∧ 𝑥 = 𝑎) → (0g‘𝑍) = (0g‘𝑍))
51		id 22	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → 𝑎 ∈ (Base‘𝑟))
52	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (0g‘𝑍) ∈ V)
53	49, 50, 51, 52	fvmptd 6936	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
54	53	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
55	48, 54	eqtr4d 2769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎))
56	36, 39, 55	eqfnfvd 6967	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
57	56	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
58	34, 57	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
59	58	alrimiv 1928	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
60	17, 29, 59	3jca 1128	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
61	16, 60	mpdan 687	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
62		eleq1 2819	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
63	62	eqeu 3665	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
64	61, 63	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
65	64	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
66	1	ringccat 20576	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
67	3, 66	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
68	2, 19, 67, 24	istermo 17901	. 2 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍)))
69	65, 68	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  Vcvv 3436   ∖ cdif 3899   ∩ cin 3901  {csn 4576   ↦ cmpt 5172   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Hom chom 17169  0_gc0g 17340  Catccat 17567  TermOctermo 17886  Ringcrg 20149   RingHom crh 20385  NzRingcnzr 20425  RingCatcringc 20558

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 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9791  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-xnn0 12452  df-z 12466  df-dec 12586  df-uz 12730  df-fz 13405  df-hash 14235  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-hom 17182  df-cco 17183  df-0g 17342  df-cat 17571  df-cid 17572  df-homf 17573  df-ssc 17714  df-resc 17715  df-subc 17716  df-termo 17889  df-estrc 18026  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-grp 18846  df-minusg 18847  df-ghm 19123  df-cmn 19692  df-abl 19693  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-rhm 20388  df-nzr 20426  df-ringc 20559

This theorem is referenced by:  nzerooringczr  21415