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Theorem nzerooringczr 44177
Description: There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
nzerooringczr.u (𝜑𝑈𝑉)
nzerooringczr.c 𝐶 = (RingCat‘𝑈)
nzerooringczr.z (𝜑𝑍 ∈ (Ring ∖ NzRing))
nzerooringczr.e (𝜑𝑍𝑈)
nzerooringczr.i (𝜑 → ℤring𝑈)
Assertion
Ref Expression
nzerooringczr (𝜑 → (ZeroO‘𝐶) = ∅)

Proof of Theorem nzerooringczr
Dummy variables 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . 2 ((ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
2 neq0 4312 . . 3 (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ ∈ (ZeroO‘𝐶))
3 nzerooringczr.u . . . . . . . 8 (𝜑𝑈𝑉)
4 nzerooringczr.c . . . . . . . . 9 𝐶 = (RingCat‘𝑈)
54ringccat 44129 . . . . . . . 8 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . . . . . 7 (𝜑𝐶 ∈ Cat)
7 iszeroi 17261 . . . . . . 7 ((𝐶 ∈ Cat ∧ ∈ (ZeroO‘𝐶)) → ( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))))
86, 7sylan 580 . . . . . 6 ((𝜑 ∈ (ZeroO‘𝐶)) → ( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))))
9 nzerooringczr.z . . . . . . . . 9 (𝜑𝑍 ∈ (Ring ∖ NzRing))
10 nzerooringczr.e . . . . . . . . 9 (𝜑𝑍𝑈)
113, 4, 9, 10zrtermoringc 44175 . . . . . . . 8 (𝜑𝑍 ∈ (TermO‘𝐶))
12 nzerooringczr.i . . . . . . . . . 10 (𝜑 → ℤring𝑈)
133, 12, 4irinitoringc 44174 . . . . . . . . 9 (𝜑 → ℤring ∈ (InitO‘𝐶))
146ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15 simplr 765 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ∈ (InitO‘𝐶))
16 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℤring ∈ (InitO‘𝐶))
1714, 15, 16initoeu1w 17264 . . . . . . . . . . . . . . . 16 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ( ≃𝑐𝐶)ℤring)
186ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝐶 ∈ Cat)
19 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍 ∈ (TermO‘𝐶))
20 simplr 765 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ∈ (TermO‘𝐶))
2118, 19, 20termoeu1w 17271 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐𝐶))
22 cictr 17067 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → 𝑍( ≃𝑐𝐶)ℤring)
236, 22syl3an1 1157 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → 𝑍( ≃𝑐𝐶)ℤring)
24 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Iso‘𝐶) = (Iso‘𝐶)
25 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Base‘𝐶) = (Base‘𝐶)
269eldifad 3951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑍 ∈ Ring)
2710, 26elind 4174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
284, 25, 3ringcbas 44116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
2927, 28eleqtrrd 2920 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑍 ∈ (Base‘𝐶))
30 zringring 20538 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ring ∈ Ring
3130a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → ℤring ∈ Ring)
3212, 31elind 4174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
3332, 28eleqtrrd 2920 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ℤring ∈ (Base‘𝐶))
3424, 25, 6, 29, 33cic 17061 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑍( ≃𝑐𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35 n0 4313 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Hom ‘𝐶) = (Hom ‘𝐶)
3725, 36, 24, 6, 29, 33isohom 17038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38 ssn0 4357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝑍(Hom ‘𝐶)ℤring) ≠ ∅)
394, 25, 3, 36, 29, 33ringchom 44118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
4039neeq1d 3079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41 zringnzr 20547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ring ∈ NzRing
42 nrhmzr 43978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
439, 41, 42sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (𝑍 RingHom ℤring) = ∅)
44 eqneqall 3031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑍 RingHom ℤring) = ∅ → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
4543, 44syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
4640, 45sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
4738, 46syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝜑 → (ZeroO‘𝐶) = ∅))
4847expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → (𝜑 → (ZeroO‘𝐶) = ∅)))
4948com13 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅)))
5037, 49mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
5135, 50syl5bir 244 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅))
5234, 51sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑍( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))
53523ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → (𝑍( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))
5423, 53mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → (ZeroO‘𝐶) = ∅)
55543exp 1113 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑍( ≃𝑐𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
5655a1dd 50 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑍( ≃𝑐𝐶) → ( ∈ (Base‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
5756ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → (𝑍( ≃𝑐𝐶) → ( ∈ (Base‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
5821, 57mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ( ∈ (Base‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
5958exp31 420 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ( ∈ (TermO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ( ∈ (Base‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
6059com34 91 . . . . . . . . . . . . . . . . . 18 (𝜑 → ( ∈ (TermO‘𝐶) → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
6160com25 99 . . . . . . . . . . . . . . . . 17 (𝜑 → (( ≃𝑐𝐶)ℤring → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ( ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6261ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → (( ≃𝑐𝐶)ℤring → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ( ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6317, 62mpd 15 . . . . . . . . . . . . . . 15 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ( ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅))))
6463ex 413 . . . . . . . . . . . . . 14 ((𝜑 ∈ (InitO‘𝐶)) → (ℤring ∈ (InitO‘𝐶) → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ( ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6564com25 99 . . . . . . . . . . . . 13 ((𝜑 ∈ (InitO‘𝐶)) → ( ∈ (TermO‘𝐶) → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6665expimpd 454 . . . . . . . . . . . 12 (𝜑 → (( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶)) → ( ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6766com23 86 . . . . . . . . . . 11 (𝜑 → ( ∈ (Base‘𝐶) → (( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶)) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
6867impd 411 . . . . . . . . . 10 (𝜑 → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅))))
6968com24 95 . . . . . . . . 9 (𝜑 → (ℤring ∈ (InitO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))))
7013, 69mpd 15 . . . . . . . 8 (𝜑 → (𝑍 ∈ (TermO‘𝐶) → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅)))
7111, 70mpd 15 . . . . . . 7 (𝜑 → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
7271adantr 481 . . . . . 6 ((𝜑 ∈ (ZeroO‘𝐶)) → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
738, 72mpd 15 . . . . 5 ((𝜑 ∈ (ZeroO‘𝐶)) → (ZeroO‘𝐶) = ∅)
7473expcom 414 . . . 4 ( ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
7574exlimiv 1924 . . 3 (∃ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
762, 75sylbi 218 . 2 (¬ (ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
771, 76pm2.61i 183 1 (𝜑 → (ZeroO‘𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3020  cdif 3936  cin 3938  wss 3939  c0 4294   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  Hom chom 16568  Catccat 16927  Isociso 17008  𝑐 ccic 17057  InitOcinito 17240  TermOctermo 17241  ZeroOczeroo 17242  Ringcrg 19219   RingHom crh 19386  NzRingcnzr 19951  ringzring 20535  RingCatcringc 44108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  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 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-seq 13363  df-hash 13684  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-0g 16707  df-cat 16931  df-cid 16932  df-homf 16933  df-sect 17009  df-inv 17010  df-iso 17011  df-cic 17058  df-ssc 17072  df-resc 17073  df-subc 17074  df-inito 17243  df-termo 17244  df-zeroo 17245  df-estrc 17365  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-grp 18038  df-minusg 18039  df-mulg 18157  df-subg 18208  df-ghm 18288  df-cmn 18830  df-mgp 19162  df-ur 19174  df-ring 19221  df-cring 19222  df-rnghom 19389  df-subrg 19455  df-nzr 19952  df-cnfld 20464  df-zring 20536  df-ringc 44110
This theorem is referenced by: (None)
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