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Theorem zrinitorngc 44554
Description: The zero ring is an initial object in the category of nonunital rings. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
zrinitorngc.u (𝜑𝑈𝑉)
zrinitorngc.c 𝐶 = (RngCat‘𝑈)
zrinitorngc.z (𝜑𝑍 ∈ (Ring ∖ NzRing))
zrinitorngc.e (𝜑𝑍𝑈)
Assertion
Ref Expression
zrinitorngc (𝜑𝑍 ∈ (InitO‘𝐶))

Proof of Theorem zrinitorngc
Dummy variables 𝑎 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zrinitorngc.c . . . . . . . . . 10 𝐶 = (RngCat‘𝑈)
2 eqid 2824 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrinitorngc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3rngcbas 44519 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
54eleq2d 2901 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6 elin 3935 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟𝑈𝑟 ∈ Rng))
76simprbi 500 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
85, 7syl6bi 256 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
98imp 410 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10 zrinitorngc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 484 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2824 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
13 eqid 2824 . . . . . . 7 (0g𝑟) = (0g𝑟)
14 eqid 2824 . . . . . . 7 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))
1512, 13, 14zrrnghm 44471 . . . . . 6 ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
169, 11, 15syl2anc 587 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
17 simpr 488 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
183adantr 484 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2824 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 zrinitorngc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
21 eldifi 4089 . . . . . . . . . . . . . 14 (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22 ringrng 44433 . . . . . . . . . . . . . 14 (𝑍 ∈ Ring → 𝑍 ∈ Rng)
2310, 21, 223syl 18 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Rng)
2420, 23elind 4156 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Rng))
2524, 4eleqtrrd 2919 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2625adantr 484 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27 simpr 488 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
281, 2, 18, 19, 26, 27rngchom 44521 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHomo 𝑟))
2928eqcomd 2830 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHomo 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
3029eleq2d 2901 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
3130biimpa 480 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
3228eleq2d 2901 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ∈ (𝑍 RngHomo 𝑟)))
33 eqid 2824 . . . . . . . . . . . . 13 (Base‘𝑟) = (Base‘𝑟)
3412, 33rnghmf 44453 . . . . . . . . . . . 12 ( ∈ (𝑍 RngHomo 𝑟) → :(Base‘𝑍)⟶(Base‘𝑟))
3532, 34syl6bi 256 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → :(Base‘𝑍)⟶(Base‘𝑟)))
3635imp 410 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → :(Base‘𝑍)⟶(Base‘𝑟))
37 ffn 6503 . . . . . . . . . . . 12 (:(Base‘𝑍)⟶(Base‘𝑟) → Fn (Base‘𝑍))
3837adantl 485 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → Fn (Base‘𝑍))
39 fvex 6674 . . . . . . . . . . . . 13 (0g𝑟) ∈ V
4039, 14fnmpti 6480 . . . . . . . . . . . 12 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) Fn (Base‘𝑍)
4140a1i 11 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) Fn (Base‘𝑍))
4232biimpa 480 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ∈ (𝑍 RngHomo 𝑟))
43 rnghmghm 44452 . . . . . . . . . . . . . 14 ( ∈ (𝑍 RngHomo 𝑟) → ∈ (𝑍 GrpHom 𝑟))
44 eqid 2824 . . . . . . . . . . . . . . 15 (0g𝑍) = (0g𝑍)
4544, 13ghmid 18364 . . . . . . . . . . . . . 14 ( ∈ (𝑍 GrpHom 𝑟) → (‘(0g𝑍)) = (0g𝑟))
4642, 43, 453syl 18 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (‘(0g𝑍)) = (0g𝑟))
4746ad2antrr 725 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (‘(0g𝑍)) = (0g𝑟))
4812, 440ringbas 44425 . . . . . . . . . . . . . . . . . 18 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4910, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
5049eleq2d 2901 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g𝑍)}))
51 elsni 4567 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {(0g𝑍)} → 𝑎 = (0g𝑍))
5251fveq2d 6665 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(0g𝑍)} → (𝑎) = (‘(0g𝑍)))
5350, 52syl6bi 256 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5453adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5554ad2antrr 725 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5655imp 410 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = (‘(0g𝑍)))
57 eqidd 2825 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
58 eqidd 2825 . . . . . . . . . . . . . 14 ((𝑎 ∈ (Base‘𝑍) ∧ 𝑥 = 𝑎) → (0g𝑟) = (0g𝑟))
59 id 22 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → 𝑎 ∈ (Base‘𝑍))
6039a1i 11 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (0g𝑟) ∈ V)
6157, 58, 59, 60fvmptd 6766 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6261adantl 485 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6347, 56, 623eqtr4d 2869 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎))
6438, 41, 63eqfnfvd 6796 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6536, 64mpdan 686 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6665ex 416 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6766adantr 484 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6867alrimiv 1929 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6917, 31, 683jca 1125 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))))
7016, 69mpdan 686 . . . 4 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))))
71 eleq1 2903 . . . . 5 ( = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
7271eqeu 3683 . . . 4 (((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))) → ∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
7370, 72syl 17 . . 3 ((𝜑𝑟 ∈ (Base‘𝐶)) → ∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
7473ralrimiva 3177 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
751rngccat 44532 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
763, 75syl 17 . . 3 (𝜑𝐶 ∈ Cat)
772, 19, 76, 25isinito 17260 . 2 (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟)))
7874, 77mpbird 260 1 (𝜑𝑍 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2115  ∃!weu 2654  wral 3133  Vcvv 3480  cdif 3916  cin 3918  {csn 4550  cmpt 5132   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  Hom chom 16576  0gc0g 16713  Catccat 16935  InitOcinito 17248   GrpHom cghm 18355  Ringcrg 19297  NzRingcnzr 20030  Rngcrng 44428   RngHomo crngh 44439  RngCatcrngc 44511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-xnn0 11965  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-hom 16589  df-cco 16590  df-0g 16715  df-cat 16939  df-cid 16940  df-homf 16941  df-ssc 17080  df-resc 17081  df-subc 17082  df-inito 17251  df-estrc 17373  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-grp 18106  df-minusg 18107  df-ghm 18356  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-nzr 20031  df-mgmhm 44329  df-rng0 44429  df-rnghomo 44441  df-rngc 44513
This theorem is referenced by:  zrzeroorngc  44556
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