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Theorem zrinitorngc 46999
Description: The zero ring is an initial object in the category of non-unital 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 2731 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrinitorngc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3rngcbas 46964 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
54eleq2d 2818 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6 elin 3964 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟𝑈𝑟 ∈ Rng))
76simprbi 496 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
85, 7syl6bi 253 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
98imp 406 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10 zrinitorngc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 480 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2731 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
13 eqid 2731 . . . . . . 7 (0g𝑟) = (0g𝑟)
14 eqid 2731 . . . . . . 7 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))
1512, 13, 14zrrnghm 20429 . . . . . 6 ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟))
169, 11, 15syl2anc 583 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟))
17 simpr 484 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟))
183adantr 480 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2731 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 zrinitorngc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
21 eldifi 4126 . . . . . . . . . . . . . 14 (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22 ringrng 20177 . . . . . . . . . . . . . 14 (𝑍 ∈ Ring → 𝑍 ∈ Rng)
2310, 21, 223syl 18 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Rng)
2420, 23elind 4194 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Rng))
2524, 4eleqtrrd 2835 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2625adantr 480 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27 simpr 484 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
281, 2, 18, 19, 26, 27rngchom 46966 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHom 𝑟))
2928eqcomd 2737 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHom 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
3029eleq2d 2818 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
3130biimpa 476 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
3228eleq2d 2818 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ∈ (𝑍 RngHom 𝑟)))
33 eqid 2731 . . . . . . . . . . . . 13 (Base‘𝑟) = (Base‘𝑟)
3412, 33rnghmf 20343 . . . . . . . . . . . 12 ( ∈ (𝑍 RngHom 𝑟) → :(Base‘𝑍)⟶(Base‘𝑟))
3532, 34syl6bi 253 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → :(Base‘𝑍)⟶(Base‘𝑟)))
3635imp 406 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → :(Base‘𝑍)⟶(Base‘𝑟))
37 ffn 6717 . . . . . . . . . . . 12 (:(Base‘𝑍)⟶(Base‘𝑟) → Fn (Base‘𝑍))
3837adantl 481 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → Fn (Base‘𝑍))
39 fvex 6904 . . . . . . . . . . . . 13 (0g𝑟) ∈ V
4039, 14fnmpti 6693 . . . . . . . . . . . 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 476 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ∈ (𝑍 RngHom 𝑟))
43 rnghmghm 20342 . . . . . . . . . . . . . 14 ( ∈ (𝑍 RngHom 𝑟) → ∈ (𝑍 GrpHom 𝑟))
44 eqid 2731 . . . . . . . . . . . . . . 15 (0g𝑍) = (0g𝑍)
4544, 13ghmid 19140 . . . . . . . . . . . . . 14 ( ∈ (𝑍 GrpHom 𝑟) → (‘(0g𝑍)) = (0g𝑟))
4642, 43, 453syl 18 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (‘(0g𝑍)) = (0g𝑟))
4746ad2antrr 723 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (‘(0g𝑍)) = (0g𝑟))
4812, 440ringbas 20421 . . . . . . . . . . . . . . . . . 18 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4910, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
5049eleq2d 2818 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g𝑍)}))
51 elsni 4645 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {(0g𝑍)} → 𝑎 = (0g𝑍))
5251fveq2d 6895 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(0g𝑍)} → (𝑎) = (‘(0g𝑍)))
5350, 52syl6bi 253 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5453adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5554ad2antrr 723 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5655imp 406 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = (‘(0g𝑍)))
57 eqidd 2732 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
58 eqidd 2732 . . . . . . . . . . . . . 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 7005 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6261adantl 481 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎) = (0g𝑟))
6347, 56, 623eqtr4d 2781 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎))
6438, 41, 63eqfnfvd 7035 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6536, 64mpdan 684 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6665ex 412 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6766adantr 480 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6867alrimiv 1929 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟)) → ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6917, 31, 683jca 1127 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))))
7016, 69mpdan 684 . . . 4 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))))
71 eleq1 2820 . . . . 5 ( = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
7271eqeu 3702 . . . 4 (((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHom 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 3145 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
751rngccat 46977 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
763, 75syl 17 . . 3 (𝜑𝐶 ∈ Cat)
772, 19, 76, 25isinito 17953 . 2 (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟)))
7874, 77mpbird 257 1 (𝜑𝑍 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1538   = wceq 1540  wcel 2105  ∃!weu 2561  wral 3060  Vcvv 3473  cdif 3945  cin 3947  {csn 4628  cmpt 5231   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  Basecbs 17151  Hom chom 17215  0gc0g 17392  Catccat 17615  InitOcinito 17938   GrpHom cghm 19131  Rngcrng 20050  Ringcrg 20131   RngHom crnghm 20329  NzRingcnzr 20407  RngCatcrngc 46956
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-xnn0 12552  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-hom 17228  df-cco 17229  df-0g 17394  df-cat 17619  df-cid 17620  df-homf 17621  df-ssc 17764  df-resc 17765  df-subc 17766  df-inito 17941  df-estrc 18081  df-mgm 18568  df-mgmhm 18620  df-sgrp 18647  df-mnd 18663  df-mhm 18708  df-grp 18861  df-minusg 18862  df-ghm 19132  df-cmn 19695  df-abl 19696  df-mgp 20033  df-rng 20051  df-ur 20080  df-ring 20133  df-rnghm 20331  df-nzr 20408  df-rngc 46958
This theorem is referenced by:  zrzeroorngc  47001
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