Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nzerooringczr Structured version   Visualization version   GIF version

Theorem nzerooringczr 42397
 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 3963 . . 3 (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ ∈ (ZeroO‘𝐶))
3 nzerooringczr.u . . . . . . . 8 (𝜑𝑈𝑉)
4 nzerooringczr.c . . . . . . . . 9 𝐶 = (RingCat‘𝑈)
54ringccat 42349 . . . . . . . 8 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . . . . . 7 (𝜑𝐶 ∈ Cat)
7 iszeroi 16706 . . . . . . 7 ((𝐶 ∈ Cat ∧ ∈ (ZeroO‘𝐶)) → ( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))))
86, 7sylan 487 . . . . . 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 42395 . . . . . . . 8 (𝜑𝑍 ∈ (TermO‘𝐶))
12 nzerooringczr.i . . . . . . . . . 10 (𝜑 → ℤring𝑈)
133, 12, 4irinitoringc 42394 . . . . . . . . 9 (𝜑 → ℤring ∈ (InitO‘𝐶))
146ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ∈ (InitO‘𝐶))
16 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℤring ∈ (InitO‘𝐶))
1714, 15, 16initoeu1w 16709 . . . . . . . . . . . . . . . 16 (((𝜑 ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ( ≃𝑐𝐶)ℤring)
186ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝐶 ∈ Cat)
19 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍 ∈ (TermO‘𝐶))
20 simplr 807 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ∈ (TermO‘𝐶))
2118, 19, 20termoeu1w 16716 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐𝐶))
22 cictr 16512 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → 𝑍( ≃𝑐𝐶)ℤring)
236, 22syl3an1 1399 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → 𝑍( ≃𝑐𝐶)ℤring)
24 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Iso‘𝐶) = (Iso‘𝐶)
25 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Base‘𝐶) = (Base‘𝐶)
269eldifad 3619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑍 ∈ Ring)
2710, 26elind 3831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
284, 25, 3ringcbas 42336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
2927, 28eleqtrrd 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑍 ∈ (Base‘𝐶))
30 zringring 19869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ring ∈ Ring
3130a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → ℤring ∈ Ring)
3212, 31elind 3831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
3332, 28eleqtrrd 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ℤring ∈ (Base‘𝐶))
3424, 25, 6, 29, 33cic 16506 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑍( ≃𝑐𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35 n0 3964 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Hom ‘𝐶) = (Hom ‘𝐶)
3725, 36, 24, 6, 29, 33isohom 16483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38 ssn0 4009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝑍(Hom ‘𝐶)ℤring) ≠ ∅)
394, 25, 3, 36, 29, 33ringchom 42338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
4039neeq1d 2882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41 zringnzr 19878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ring ∈ NzRing
42 nrhmzr 42198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
439, 41, 42sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (𝑍 RingHom ℤring) = ∅)
44 eqneqall 2834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑍 RingHom ℤring) = ∅ → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
4543, 44syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
4640, 45sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅))
5234, 51sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑍( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))
53523ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → (𝑍( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))
5423, 53mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑍( ≃𝑐𝐶)( ≃𝑐𝐶)ℤring) → (ZeroO‘𝐶) = ∅)
55543exp 1283 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑍( ≃𝑐𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
5655a1dd 50 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑍( ≃𝑐𝐶) → ( ∈ (Base‘𝐶) → (( ≃𝑐𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
5756ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 628 . . . . . . . . . . . 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 446 . . . . . . . . . 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 480 . . . . . 6 ((𝜑 ∈ (ZeroO‘𝐶)) → (( ∈ (Base‘𝐶) ∧ ( ∈ (InitO‘𝐶) ∧ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
738, 72mpd 15 . . . . 5 ((𝜑 ∈ (ZeroO‘𝐶)) → (ZeroO‘𝐶) = ∅)
7473expcom 450 . . . 4 ( ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
7574exlimiv 1898 . . 3 (∃ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
762, 75sylbi 207 . 2 (¬ (ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
771, 76pm2.61i 176 1 (𝜑 → (ZeroO‘𝐶) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  Catccat 16372  Isociso 16453   ≃𝑐 ccic 16502  InitOcinito 16685  TermOctermo 16686  ZeroOczeroo 16687  Ringcrg 18593   RingHom crh 18760  NzRingcnzr 19305  ℤringzring 19866  RingCatcringc 42328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-0g 16149  df-cat 16376  df-cid 16377  df-homf 16378  df-sect 16454  df-inv 16455  df-iso 16456  df-cic 16503  df-ssc 16517  df-resc 16518  df-subc 16519  df-inito 16688  df-termo 16689  df-zeroo 16690  df-estrc 16810  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-minusg 17473  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cmn 18241  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-rnghom 18763  df-subrg 18826  df-nzr 19306  df-cnfld 19795  df-zring 19867  df-ringc 42330 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator