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Theorem zrinitorngc 44199
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 2818 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrinitorngc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3rngcbas 44164 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
54eleq2d 2895 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6 elin 4166 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟𝑈𝑟 ∈ Rng))
76simprbi 497 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
85, 7syl6bi 254 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
98imp 407 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10 zrinitorngc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 481 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2818 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
13 eqid 2818 . . . . . . 7 (0g𝑟) = (0g𝑟)
14 eqid 2818 . . . . . . 7 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))
1512, 13, 14zrrnghm 44116 . . . . . 6 ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
169, 11, 15syl2anc 584 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
17 simpr 485 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
183adantr 481 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2818 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 zrinitorngc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
21 eldifi 4100 . . . . . . . . . . . . . 14 (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22 ringrng 44078 . . . . . . . . . . . . . 14 (𝑍 ∈ Ring → 𝑍 ∈ Rng)
2310, 21, 223syl 18 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Rng)
2420, 23elind 4168 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Rng))
2524, 4eleqtrrd 2913 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2625adantr 481 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27 simpr 485 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
281, 2, 18, 19, 26, 27rngchom 44166 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHomo 𝑟))
2928eqcomd 2824 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHomo 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
3029eleq2d 2895 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
3130biimpa 477 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
3228eleq2d 2895 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ∈ (𝑍 RngHomo 𝑟)))
33 eqid 2818 . . . . . . . . . . . . 13 (Base‘𝑟) = (Base‘𝑟)
3412, 33rnghmf 44098 . . . . . . . . . . . 12 ( ∈ (𝑍 RngHomo 𝑟) → :(Base‘𝑍)⟶(Base‘𝑟))
3532, 34syl6bi 254 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → :(Base‘𝑍)⟶(Base‘𝑟)))
3635imp 407 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → :(Base‘𝑍)⟶(Base‘𝑟))
37 ffn 6507 . . . . . . . . . . . 12 (:(Base‘𝑍)⟶(Base‘𝑟) → Fn (Base‘𝑍))
3837adantl 482 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → Fn (Base‘𝑍))
39 fvex 6676 . . . . . . . . . . . . 13 (0g𝑟) ∈ V
4039, 14fnmpti 6484 . . . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ∈ (𝑍 RngHomo 𝑟))
43 rnghmghm 44097 . . . . . . . . . . . . . 14 ( ∈ (𝑍 RngHomo 𝑟) → ∈ (𝑍 GrpHom 𝑟))
44 eqid 2818 . . . . . . . . . . . . . . 15 (0g𝑍) = (0g𝑍)
4544, 13ghmid 18302 . . . . . . . . . . . . . 14 ( ∈ (𝑍 GrpHom 𝑟) → (‘(0g𝑍)) = (0g𝑟))
4642, 43, 453syl 18 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (‘(0g𝑍)) = (0g𝑟))
4746ad2antrr 722 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (‘(0g𝑍)) = (0g𝑟))
4812, 440ringbas 44070 . . . . . . . . . . . . . . . . . 18 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4910, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
5049eleq2d 2895 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g𝑍)}))
51 elsni 4574 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {(0g𝑍)} → 𝑎 = (0g𝑍))
5251fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(0g𝑍)} → (𝑎) = (‘(0g𝑍)))
5350, 52syl6bi 254 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5453adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5554ad2antrr 722 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5655imp 407 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = (‘(0g𝑍)))
57 eqidd 2819 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
58 eqidd 2819 . . . . . . . . . . . . . 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 6767 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6261adantl 482 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6347, 56, 623eqtr4d 2863 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎))
6438, 41, 63eqfnfvd 6797 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6536, 64mpdan 683 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6665ex 413 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6766adantr 481 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6867alrimiv 1919 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6917, 31, 683jca 1120 . . . . 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 683 . . . 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 2897 . . . . 5 ( = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
7271eqeu 3694 . . . 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 3179 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
751rngccat 44177 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
763, 75syl 17 . . 3 (𝜑𝐶 ∈ Cat)
772, 19, 76, 25isinito 17248 . 2 (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟)))
7874, 77mpbird 258 1 (𝜑𝑍 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  ∃!weu 2646  wral 3135  Vcvv 3492  cdif 3930  cin 3932  {csn 4557  cmpt 5137   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  0gc0g 16701  Catccat 16923  InitOcinito 17236   GrpHom cghm 18293  Ringcrg 19226  NzRingcnzr 19958  Rngcrng 44073   RngHomo crngh 44084  RngCatcrngc 44156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-hash 13679  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-hom 16577  df-cco 16578  df-0g 16703  df-cat 16927  df-cid 16928  df-homf 16929  df-ssc 17068  df-resc 17069  df-subc 17070  df-inito 17239  df-estrc 17361  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-grp 18044  df-minusg 18045  df-ghm 18294  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-nzr 19959  df-mgmhm 43923  df-rng0 44074  df-rnghomo 44086  df-rngc 44158
This theorem is referenced by:  zrzeroorngc  44201
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