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Theorem zrinitorngc 45527
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 2740 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrinitorngc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3rngcbas 45492 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
54eleq2d 2826 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6 elin 3908 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟𝑈𝑟 ∈ Rng))
76simprbi 497 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
85, 7syl6bi 252 . . . . . . 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 2740 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
13 eqid 2740 . . . . . . 7 (0g𝑟) = (0g𝑟)
14 eqid 2740 . . . . . . 7 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))
1512, 13, 14zrrnghm 45444 . . . . . 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 2740 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 zrinitorngc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
21 eldifi 4066 . . . . . . . . . . . . . 14 (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22 ringrng 45406 . . . . . . . . . . . . . 14 (𝑍 ∈ Ring → 𝑍 ∈ Rng)
2310, 21, 223syl 18 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Rng)
2420, 23elind 4133 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Rng))
2524, 4eleqtrrd 2844 . . . . . . . . . . 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 45494 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHomo 𝑟))
2928eqcomd 2746 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHomo 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
3029eleq2d 2826 . . . . . . 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 2826 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ∈ (𝑍 RngHomo 𝑟)))
33 eqid 2740 . . . . . . . . . . . . 13 (Base‘𝑟) = (Base‘𝑟)
3412, 33rnghmf 45426 . . . . . . . . . . . 12 ( ∈ (𝑍 RngHomo 𝑟) → :(Base‘𝑍)⟶(Base‘𝑟))
3532, 34syl6bi 252 . . . . . . . . . . 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 6598 . . . . . . . . . . . 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 6784 . . . . . . . . . . . . 13 (0g𝑟) ∈ V
4039, 14fnmpti 6574 . . . . . . . . . . . 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 45425 . . . . . . . . . . . . . 14 ( ∈ (𝑍 RngHomo 𝑟) → ∈ (𝑍 GrpHom 𝑟))
44 eqid 2740 . . . . . . . . . . . . . . 15 (0g𝑍) = (0g𝑍)
4544, 13ghmid 18838 . . . . . . . . . . . . . 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 45398 . . . . . . . . . . . . . . . . . 18 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4910, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
5049eleq2d 2826 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g𝑍)}))
51 elsni 4584 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {(0g𝑍)} → 𝑎 = (0g𝑍))
5251fveq2d 6775 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(0g𝑍)} → (𝑎) = (‘(0g𝑍)))
5350, 52syl6bi 252 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5453adantr 481 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = (‘(0g𝑍)))
57 eqidd 2741 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
58 eqidd 2741 . . . . . . . . . . . . . 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 6879 . . . . . . . . . . . . 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 2790 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎))
6438, 41, 63eqfnfvd 6909 . . . . . . . . . 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 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 1934 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6917, 31, 683jca 1127 . . . . 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 684 . . . 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 2828 . . . . 5 ( = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
7271eqeu 3645 . . . 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 3110 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
751rngccat 45505 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
763, 75syl 17 . . 3 (𝜑𝐶 ∈ Cat)
772, 19, 76, 25isinito 17709 . 2 (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟)))
7874, 77mpbird 256 1 (𝜑𝑍 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wal 1540   = wceq 1542  wcel 2110  ∃!weu 2570  wral 3066  Vcvv 3431  cdif 3889  cin 3891  {csn 4567  cmpt 5162   Fn wfn 6427  wf 6428  cfv 6432  (class class class)co 7271  Basecbs 16910  Hom chom 16971  0gc0g 17148  Catccat 17371  InitOcinito 17694   GrpHom cghm 18829  Ringcrg 19781  NzRingcnzr 20526  Rngcrng 45401   RngHomo crngh 45412  RngCatcrngc 45484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-oadd 8292  df-er 8481  df-map 8600  df-pm 8601  df-ixp 8669  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-dju 9660  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  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 12437  df-uz 12582  df-fz 13239  df-hash 14043  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-hom 16984  df-cco 16985  df-0g 17150  df-cat 17375  df-cid 17376  df-homf 17377  df-ssc 17520  df-resc 17521  df-subc 17522  df-inito 17697  df-estrc 17837  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-mhm 18428  df-grp 18578  df-minusg 18579  df-ghm 18830  df-cmn 19386  df-abl 19387  df-mgp 19719  df-ur 19736  df-ring 19783  df-nzr 20527  df-mgmhm 45302  df-rng0 45402  df-rnghomo 45414  df-rngc 45486
This theorem is referenced by:  zrzeroorngc  45529
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