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Theorem zrinitorngc 41314
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 2621 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrinitorngc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3rngcbas 41279 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
54eleq2d 2684 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6 elin 3779 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟𝑈𝑟 ∈ Rng))
76simprbi 480 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
85, 7syl6bi 243 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
98imp 445 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10 zrinitorngc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 481 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2621 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
13 eqid 2621 . . . . . . 7 (0g𝑟) = (0g𝑟)
14 eqid 2621 . . . . . . 7 (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))
1512, 13, 14zrrnghm 41231 . . . . . 6 ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
169, 11, 15syl2anc 692 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟))
17 simpr 477 . . . . . 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 2621 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 zrinitorngc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
21 eldifi 3715 . . . . . . . . . . . . . 14 (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22 ringrng 41193 . . . . . . . . . . . . . 14 (𝑍 ∈ Ring → 𝑍 ∈ Rng)
2310, 21, 223syl 18 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Rng)
2420, 23elind 3781 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Rng))
2524, 4eleqtrrd 2701 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2625adantr 481 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27 simpr 477 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
281, 2, 18, 19, 26, 27rngchom 41281 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHomo 𝑟))
2928eqcomd 2627 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHomo 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
3029eleq2d 2684 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
3130biimpa 501 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
3228eleq2d 2684 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ∈ (𝑍 RngHomo 𝑟)))
33 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑟) = (Base‘𝑟)
3412, 33rnghmf 41213 . . . . . . . . . . . 12 ( ∈ (𝑍 RngHomo 𝑟) → :(Base‘𝑍)⟶(Base‘𝑟))
3532, 34syl6bi 243 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) → :(Base‘𝑍)⟶(Base‘𝑟)))
3635imp 445 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → :(Base‘𝑍)⟶(Base‘𝑟))
37 ffn 6007 . . . . . . . . . . . 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 6163 . . . . . . . . . . . . 13 (0g𝑟) ∈ V
4039, 14fnmpti 5984 . . . . . . . . . . . 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 501 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ∈ (𝑍 RngHomo 𝑟))
43 rnghmghm 41212 . . . . . . . . . . . . . 14 ( ∈ (𝑍 RngHomo 𝑟) → ∈ (𝑍 GrpHom 𝑟))
44 eqid 2621 . . . . . . . . . . . . . . 15 (0g𝑍) = (0g𝑍)
4544, 13ghmid 17598 . . . . . . . . . . . . . 14 ( ∈ (𝑍 GrpHom 𝑟) → (‘(0g𝑍)) = (0g𝑟))
4642, 43, 453syl 18 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (‘(0g𝑍)) = (0g𝑟))
4746ad2antrr 761 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (‘(0g𝑍)) = (0g𝑟))
4812, 440ringbas 41185 . . . . . . . . . . . . . . . . . 18 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4910, 48syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
5049eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g𝑍)}))
51 elsni 4170 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {(0g𝑍)} → 𝑎 = (0g𝑍))
5251fveq2d 6157 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(0g𝑍)} → (𝑎) = (‘(0g𝑍)))
5350, 52syl6bi 243 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5453adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5554ad2antrr 761 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (𝑎) = (‘(0g𝑍))))
5655imp 445 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = (‘(0g𝑍)))
57 eqidd 2622 . . . . . . . . . . . . . 14 (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
58 eqidd 2622 . . . . . . . . . . . . . 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 6250 . . . . . . . . . . . . 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 2665 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))‘𝑎))
6438, 41, 63eqfnfvd 6275 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ :(Base‘𝑍)⟶(Base‘𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6536, 64mpdan 701 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ ∈ (𝑍(Hom ‘𝐶)𝑟)) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)))
6665ex 450 . . . . . . . 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 1852 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ∀( ∈ (𝑍(Hom ‘𝐶)𝑟) → = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟))))
6917, 31, 683jca 1240 . . . . 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 701 . . . 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 2686 . . . . 5 ( = (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) → ( ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
7271eqeu 3363 . . . 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 2961 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟))
751rngccat 41292 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
763, 75syl 17 . . 3 (𝜑𝐶 ∈ Cat)
772, 19, 76, 25isinito 16582 . 2 (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑍(Hom ‘𝐶)𝑟)))
7874, 77mpbird 247 1 (𝜑𝑍 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  ∃!weu 2469  wral 2907  Vcvv 3189  cdif 3556  cin 3558  {csn 4153  cmpt 4678   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  Basecbs 15792  Hom chom 15884  0gc0g 16032  Catccat 16257  InitOcinito 16570   GrpHom cghm 17589  Ringcrg 18479  NzRingcnzr 19189  Rngcrng 41188   RngHomo crngh 41199  RngCatcrngc 41271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-xnn0 11316  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-hash 13066  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-hom 15898  df-cco 15899  df-0g 16034  df-cat 16261  df-cid 16262  df-homf 16263  df-ssc 16402  df-resc 16403  df-subc 16404  df-inito 16573  df-estrc 16695  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-grp 17357  df-minusg 17358  df-ghm 17590  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-nzr 19190  df-mgmhm 41093  df-rng0 41189  df-rnghomo 41201  df-rngc 41273
This theorem is referenced by:  zrzeroorngc  41316
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