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Theorem zrtermoringc 42917
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 2825 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrtermoringc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3ringcbas 42858 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
54eleq2d 2892 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6 elin 4023 . . . . . . . . 9 (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟𝑈𝑟 ∈ Ring))
76simprbi 492 . . . . . . . 8 (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
85, 7syl6bi 245 . . . . . . 7 (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
98imp 397 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10 zrtermoringc.z . . . . . . 7 (𝜑𝑍 ∈ (Ring ∖ NzRing))
1110adantr 474 . . . . . 6 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12 eqid 2825 . . . . . . 7 (Base‘𝑟) = (Base‘𝑟)
13 eqid 2825 . . . . . . 7 (0g𝑍) = (0g𝑍)
14 eqid 2825 . . . . . . 7 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))
1512, 13, 14c0rhm 42759 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
169, 11, 15syl2anc 581 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
17 simpr 479 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍))
183adantr 474 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑈𝑉)
19 eqid 2825 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 simpr 479 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21 zrtermoringc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
2210eldifad 3810 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Ring)
2321, 22elind 4025 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
2423, 4eleqtrrd 2909 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2524adantr 474 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
261, 2, 18, 19, 20, 25ringchom 42860 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
2726eqcomd 2831 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
2827eleq2d 2892 . . . . . . 7 ((𝜑𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
2928biimpa 470 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
3026eleq2d 2892 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ∈ (𝑟 RingHom 𝑍)))
31 eqid 2825 . . . . . . . . . . 11 (Base‘𝑍) = (Base‘𝑍)
3212, 31rhmf 19082 . . . . . . . . . 10 ( ∈ (𝑟 RingHom 𝑍) → :(Base‘𝑟)⟶(Base‘𝑍))
3330, 32syl6bi 245 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
3433adantr 474 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) → :(Base‘𝑟)⟶(Base‘𝑍)))
35 ffn 6278 . . . . . . . . . . 11 (:(Base‘𝑟)⟶(Base‘𝑍) → Fn (Base‘𝑟))
3635adantl 475 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → Fn (Base‘𝑟))
37 fvex 6446 . . . . . . . . . . . 12 (0g𝑍) ∈ V
3837, 14fnmpti 6255 . . . . . . . . . . 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 42718 . . . . . . . . . . . . . . . . 17 (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g𝑍)})
4110, 40syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝑍) = {(0g𝑍)})
4241adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g𝑍)})
4342feq3d 6265 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) ↔ :(Base‘𝑟)⟶{(0g𝑍)}))
44 fvconst 6682 . . . . . . . . . . . . . . 15 ((:(Base‘𝑟)⟶{(0g𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
4544ex 403 . . . . . . . . . . . . . 14 (:(Base‘𝑟)⟶{(0g𝑍)} → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍)))
4643, 45syl6bi 245 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ (Base‘𝐶)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4746adantr 474 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → (:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (𝑎) = (0g𝑍))))
4847imp31 410 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (𝑎) = (0g𝑍))
49 eqidd 2826 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
50 eqidd 2826 . . . . . . . . . . . . 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 6535 . . . . . . . . . . . 12 (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))‘𝑎) = (0g𝑍))
5453adantl 475 . . . . . . . . . . 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 2864 . . . . . . . . . 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 6563 . . . . . . . . 9 ((((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ :(Base‘𝑟)⟶(Base‘𝑍)) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
5756ex 403 . . . . . . . 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 2028 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
6017, 29, 593jca 1164 . . . . 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 680 . . . 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 2894 . . . . 5 ( = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
6362eqeu 3600 . . . 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 3175 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
661ringccat 42871 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
673, 66syl 17 . . 3 (𝜑𝐶 ∈ Cat)
682, 19, 67, 24istermo 17003 . 2 (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍)))
6965, 68mpbird 249 1 (𝜑𝑍 ∈ (TermO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113  wal 1656   = wceq 1658  wcel 2166  ∃!weu 2639  wral 3117  Vcvv 3414  cdif 3795  cin 3797  {csn 4397  cmpt 4952   Fn wfn 6118  wf 6119  cfv 6123  (class class class)co 6905  Basecbs 16222  Hom chom 16316  0gc0g 16453  Catccat 16677  TermOctermo 16991  Ringcrg 18901   RingHom crh 19068  NzRingcnzr 19618  RingCatcringc 42850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-xnn0 11691  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-hash 13411  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-hom 16329  df-cco 16330  df-0g 16455  df-cat 16681  df-cid 16682  df-homf 16683  df-ssc 16822  df-resc 16823  df-subc 16824  df-termo 16994  df-estrc 17115  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-grp 17779  df-minusg 17780  df-ghm 18009  df-mgp 18844  df-ur 18856  df-ring 18903  df-rnghom 19071  df-nzr 19619  df-ringc 42852
This theorem is referenced by:  nzerooringczr  42919
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