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Theorem zrtermoringc 44348
Description: The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
zrtermoringc.u (𝜑𝑈𝑉)
zrtermoringc.c 𝐶 = (RingCat‘𝑈)
zrtermoringc.z (𝜑𝑍 ∈ (Ring ∖ NzRing))
zrtermoringc.e (𝜑𝑍𝑈)
Assertion
Ref Expression
zrtermoringc (𝜑𝑍 ∈ (TermO‘𝐶))

Proof of Theorem zrtermoringc
Dummy variables 𝑎 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zrtermoringc.c . . . . . . . . . 10 𝐶 = (RingCat‘𝑈)
2 eqid 2824 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrtermoringc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3ringcbas 44289 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
54eleq2d 2901 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6 elin 4172 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟𝑈𝑟 ∈ Ring))
76simprbi 499 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
85, 7syl6bi 255 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
98imp 409 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 483 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2824 . . . . . . 7 (Base‘𝑟) = (Base‘𝑟)
13 eqid 2824 . . . . . . 7 (0g𝑍) = (0g𝑍)
14 eqid 2824 . . . . . . 7 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))
1512, 13, 14c0rhm 44190 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
169, 11, 15syl2anc 586 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
17 simpr 487 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
183adantr 483 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2824 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 simpr 487 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21 zrtermoringc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
2210eldifad 3951 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Ring)
2321, 22elind 4174 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
2423, 4eleqtrrd 2919 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2524adantr 483 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
261, 2, 18, 19, 20, 25ringchom 44291 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
2726eqcomd 2830 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
2827eleq2d 2901 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
2928biimpa 479 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
3026eleq2d 2901 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ∈ (𝑟 RingHom 𝑍)))
31 eqid 2824 . . . . . . . . . . 11 (Base‘𝑍) = (Base‘𝑍)
3212, 31rhmf 19481 . . . . . . . . . 10 ( ∈ (𝑟 RingHom 𝑍) → :(Base‘𝑟)⟶(Base‘𝑍))
3330, 32syl6bi 255 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
3433adantr 483 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
35 ffn 6517 . . . . . . . . . . 11 (:(Base‘𝑟)⟶(Base‘𝑍) → Fn (Base‘𝑟))
3635adantl 484 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → Fn (Base‘𝑟))
37 fvex 6686 . . . . . . . . . . . 12 (0g𝑍) ∈ V
3837, 14fnmpti 6494 . . . . . . . . . . 11 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) Fn (Base‘𝑟)
3938a1i 11 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) Fn (Base‘𝑟))
4031, 130ringbas 44149 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
4241adantr 483 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g𝑍)})
4342feq3d 6504 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) ↔ :(Base‘𝑟)⟶{(0g𝑍)}))
44 fvconst 6929 . . . . . . . . . . . . . . 15 ((:(Base‘𝑟)⟶{(0g𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
4544ex 415 . . . . . . . . . . . . . 14 (:(Base‘𝑟)⟶{(0g𝑍)} → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍)))
4643, 45syl6bi 255 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4746adantr 483 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4847imp31 420 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
49 eqidd 2825 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
50 eqidd 2825 . . . . . . . . . . . . 13 ((𝑎 ∈ (Base‘𝑟) ∧ 𝑥 = 𝑎) → (0g𝑍) = (0g𝑍))
51 id 22 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → 𝑎 ∈ (Base‘𝑟))
5237a1i 11 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (0g𝑍) ∈ V)
5349, 50, 51, 52fvmptd 6778 . . . . . . . . . . . 12 (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎) = (0g𝑍))
5453adantl 484 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎) = (0g𝑍))
5548, 54eqtr4d 2862 . . . . . . . . . 10 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎))
5636, 39, 55eqfnfvd 6808 . . . . . . . . 9 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
5756ex 415 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (:(Base‘𝑟)⟶(Base‘𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
5834, 57syld 47 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
5958alrimiv 1927 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
6017, 29, 593jca 1124 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))))
6116, 60mpdan 685 . . . 4 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))))
62 eleq1 2903 . . . . 5 ( = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
6362eqeu 3700 . . . 4 (((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))) → ∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
6461, 63syl 17 . . 3 ((𝜑𝑟 ∈ (Base‘𝐶)) → ∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
6564ralrimiva 3185 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
661ringccat 44302 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
673, 66syl 17 . . 3 (𝜑𝐶 ∈ Cat)
682, 19, 67, 24istermo 17264 . 2 (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍)))
6965, 68mpbird 259 1 (𝜑𝑍 ∈ (TermO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wal 1534   = wceq 1536  wcel 2113  ∃!weu 2652  wral 3141  Vcvv 3497  cdif 3936  cin 3938  {csn 4570  cmpt 5149   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  Basecbs 16486  Hom chom 16579  0gc0g 16716  Catccat 16938  TermOctermo 17252  Ringcrg 19300   RingHom crh 19467  NzRingcnzr 20033  RingCatcringc 44281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-hash 13694  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-hom 16592  df-cco 16593  df-0g 16718  df-cat 16942  df-cid 16943  df-homf 16944  df-ssc 17083  df-resc 17084  df-subc 17085  df-termo 17255  df-estrc 17376  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-grp 18109  df-minusg 18110  df-ghm 18359  df-mgp 19243  df-ur 19255  df-ring 19302  df-rnghom 19470  df-nzr 20034  df-ringc 44283
This theorem is referenced by:  nzerooringczr  44350
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