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Theorem zrtermoringc 20675
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 2737 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrtermoringc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3ringcbas 20650 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
54eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6 elin 3967 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟𝑈𝑟 ∈ Ring))
76simprbi 496 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
85, 7biimtrdi 253 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
98imp 406 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 480 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2737 . . . . . . 7 (Base‘𝑟) = (Base‘𝑟)
13 eqid 2737 . . . . . . 7 (0g𝑍) = (0g𝑍)
14 eqid 2737 . . . . . . 7 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))
1512, 13, 14c0rhm 20534 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
169, 11, 15syl2anc 584 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
17 simpr 484 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
183adantr 480 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2737 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 simpr 484 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21 zrtermoringc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
2210eldifad 3963 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Ring)
2321, 22elind 4200 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
2423, 4eleqtrrd 2844 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2524adantr 480 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
261, 2, 18, 19, 20, 25ringchom 20652 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
2726eqcomd 2743 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
2827eleq2d 2827 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
2928biimpa 476 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
3026eleq2d 2827 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ∈ (𝑟 RingHom 𝑍)))
31 eqid 2737 . . . . . . . . . . 11 (Base‘𝑍) = (Base‘𝑍)
3212, 31rhmf 20485 . . . . . . . . . 10 ( ∈ (𝑟 RingHom 𝑍) → :(Base‘𝑟)⟶(Base‘𝑍))
3330, 32biimtrdi 253 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
3433adantr 480 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
35 ffn 6736 . . . . . . . . . . 11 (:(Base‘𝑟)⟶(Base‘𝑍) → Fn (Base‘𝑟))
3635adantl 481 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → Fn (Base‘𝑟))
37 fvex 6919 . . . . . . . . . . . 12 (0g𝑍) ∈ V
3837, 14fnmpti 6711 . . . . . . . . . . 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 20528 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
4241adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g𝑍)})
4342feq3d 6723 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) ↔ :(Base‘𝑟)⟶{(0g𝑍)}))
44 fvconst 7184 . . . . . . . . . . . . . . 15 ((:(Base‘𝑟)⟶{(0g𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
4544ex 412 . . . . . . . . . . . . . 14 (:(Base‘𝑟)⟶{(0g𝑍)} → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍)))
4643, 45biimtrdi 253 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4746adantr 480 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4847imp31 417 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
49 eqidd 2738 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
50 eqidd 2738 . . . . . . . . . . . . 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 7023 . . . . . . . . . . . 12 (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎) = (0g𝑍))
5453adantl 481 . . . . . . . . . . 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 2780 . . . . . . . . . 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 7054 . . . . . . . . 9 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
5756ex 412 . . . . . . . 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 1129 . . . . 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 687 . . . 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 2829 . . . . 5 ( = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
6362eqeu 3712 . . . 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 3146 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
661ringccat 20663 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
673, 66syl 17 . . 3 (𝜑𝐶 ∈ Cat)
682, 19, 67, 24istermo 18042 . 2 (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍)))
6965, 68mpbird 257 1 (𝜑𝑍 ∈ (TermO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  ∃!weu 2568  wral 3061  Vcvv 3480  cdif 3948  cin 3950  {csn 4626  cmpt 5225   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  0gc0g 17484  Catccat 17707  TermOctermo 18027  Ringcrg 20230   RingHom crh 20469  NzRingcnzr 20512  RingCatcringc 20645
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-hom 17321  df-cco 17322  df-0g 17486  df-cat 17711  df-cid 17712  df-homf 17713  df-ssc 17854  df-resc 17855  df-subc 17856  df-termo 18030  df-estrc 18167  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-rhm 20472  df-nzr 20513  df-ringc 20646
This theorem is referenced by:  nzerooringczr  21491
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