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

Theorem zrtermoringc 41358
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 2621 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrtermoringc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3ringcbas 41299 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
54eleq2d 2684 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6 elin 3774 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟𝑈𝑟 ∈ Ring))
76simprbi 480 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
85, 7syl6bi 243 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
98imp 445 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 481 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2621 . . . . . . 7 (Base‘𝑟) = (Base‘𝑟)
13 eqid 2621 . . . . . . 7 (0g𝑍) = (0g𝑍)
14 eqid 2621 . . . . . . 7 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))
1512, 13, 14c0rhm 41200 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
169, 11, 15syl2anc 692 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
17 simpr 477 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
183adantr 481 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2621 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 simpr 477 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21 zrtermoringc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
2210eldifad 3567 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Ring)
2321, 22elind 3776 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
2423, 4eleqtrrd 2701 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2524adantr 481 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
261, 2, 18, 19, 20, 25ringchom 41301 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
2726eqcomd 2627 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
2827eleq2d 2684 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
2928biimpa 501 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
3026eleq2d 2684 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ∈ (𝑟 RingHom 𝑍)))
31 eqid 2621 . . . . . . . . . . 11 (Base‘𝑍) = (Base‘𝑍)
3212, 31rhmf 18647 . . . . . . . . . 10 ( ∈ (𝑟 RingHom 𝑍) → :(Base‘𝑟)⟶(Base‘𝑍))
3330, 32syl6bi 243 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
3433adantr 481 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
35 ffn 6002 . . . . . . . . . . 11 (:(Base‘𝑟)⟶(Base‘𝑍) → Fn (Base‘𝑟))
3635adantl 482 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → Fn (Base‘𝑟))
37 fvex 6158 . . . . . . . . . . . 12 (0g𝑍) ∈ V
3837, 14fnmpti 5979 . . . . . . . . . . 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 41159 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
4241adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g𝑍)})
4342feq3d 5989 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) ↔ :(Base‘𝑟)⟶{(0g𝑍)}))
44 fvconst 6385 . . . . . . . . . . . . . . 15 ((:(Base‘𝑟)⟶{(0g𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
4544ex 450 . . . . . . . . . . . . . 14 (:(Base‘𝑟)⟶{(0g𝑍)} → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍)))
4643, 45syl6bi 243 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4746adantr 481 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4847imp31 448 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
49 eqidd 2622 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
50 eqidd 2622 . . . . . . . . . . . . 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 6245 . . . . . . . . . . . 12 (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎) = (0g𝑍))
5453adantl 482 . . . . . . . . . . 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 2658 . . . . . . . . . 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 6270 . . . . . . . . 9 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
5756ex 450 . . . . . . . 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 1852 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
6017, 29, 593jca 1240 . . . . 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 701 . . . 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 2686 . . . . 5 ( = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
6362eqeu 3359 . . . 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 2960 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
661ringccat 41312 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
673, 66syl 17 . . 3 (𝜑𝐶 ∈ Cat)
682, 19, 67, 24istermo 16572 . 2 (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍)))
6965, 68mpbird 247 1 (𝜑𝑍 ∈ (TermO‘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 3186  cdif 3552  cin 3554  {csn 4148  cmpt 4673   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  0gc0g 16021  Catccat 16246  TermOctermo 16560  Ringcrg 18468   RingHom crh 18633  NzRingcnzr 19176  RingCatcringc 41291
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-hash 13058  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-hom 15887  df-cco 15888  df-0g 16023  df-cat 16250  df-cid 16251  df-homf 16252  df-ssc 16391  df-resc 16392  df-subc 16393  df-termo 16563  df-estrc 16684  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-minusg 17347  df-ghm 17579  df-mgp 18411  df-ur 18423  df-ring 18470  df-rnghom 18636  df-nzr 19177  df-ringc 41293
This theorem is referenced by:  nzerooringczr  41360
  Copyright terms: Public domain W3C validator