Theorem zrtermoringc 45628

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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrtermoringc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	ringcbas 45569	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
5	4	eleq2d 2824	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6		elin 3903	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Ring))
7	6	simprbi 497	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
8	5, 7	syl6bi 252	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
9	8	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10		zrtermoringc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑟) = (Base‘𝑟)
13		eqid 2738	. . . . . . 7 ⊢ (0g‘𝑍) = (0g‘𝑍)
14		eqid 2738	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))
15	12, 13, 14	c0rhm 45470	. . . . . 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 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
18	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21		zrtermoringc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	10	eldifad 3899	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
23	21, 22	elind 4128	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
24	23, 4	eleqtrrd 2842	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
26	1, 2, 18, 19, 20, 25	ringchom 45571	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
27	26	eqcomd 2744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
28	27	eleq2d 2824	. . . . . . 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 477	. . . . . 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 2824	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ℎ ∈ (𝑟 RingHom 𝑍)))
31		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	12, 31	rhmf 19970	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑟 RingHom 𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍))
33	30, 32	syl6bi 252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
34	33	adantr 481	. . . . . . . 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 6600	. . . . . . . . . . 11 ⊢ (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ Fn (Base‘𝑟))
36	35	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ Fn (Base‘𝑟))
37		fvex 6787	. . . . . . . . . . . 12 ⊢ (0g‘𝑍) ∈ V
38	37, 14	fnmpti 6576	. . . . . . . . . . 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 45429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
41	10, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
42	41	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g‘𝑍)})
43	42	feq3d 6587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) ↔ ℎ:(Base‘𝑟)⟶{(0g‘𝑍)}))
44		fvconst 7036	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
45	44	ex 413	. . . . . . . . . . . . . 14 ⊢ (ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍)))
46	43, 45	syl6bi 252	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
47	46	adantr 481	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 2739	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
50		eqidd 2739	. . . . . . . . . . . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
54	53	adantl 482	. . . . . . . . . . 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 2781	. . . . . . . . . 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 6912	. . . . . . . . 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 413	. . . . . . . 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 1930	. . . . . 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 1127	. . . . 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 684	. . . 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 2826	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
63	62	eqeu 3641	. . . 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 3103	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
66	1	ringccat 45582	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
67	3, 66	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
68	2, 19, 67, 24	istermo 17712	. 2 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍)))
69	65, 68	mpbird 256	1 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886  {csn 4561   ↦ cmpt 5157   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  0_gc0g 17150  Catccat 17373  TermOctermo 17697  Ringcrg 19783   RingHom crh 19956  NzRingcnzr 20528  RingCatcringc 45561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-hom 16986  df-cco 16987  df-0g 17152  df-cat 17377  df-cid 17378  df-homf 17379  df-ssc 17522  df-resc 17523  df-subc 17524  df-termo 17700  df-estrc 17839  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-minusg 18581  df-ghm 18832  df-mgp 19721  df-ur 19738  df-ring 19785  df-rnghom 19959  df-nzr 20529  df-ringc 45563

This theorem is referenced by:  nzerooringczr  45630