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Theorem zrtermoringc 20692
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 2735 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
3 zrtermoringc.u . . . . . . . . . 10 (𝜑𝑈𝑉)
41, 2, 3ringcbas 20667 . . . . . . . . 9 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
54eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6 elin 3979 . . . . . . . . 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 2735 . . . . . . 7 (Base‘𝑟) = (Base‘𝑟)
13 eqid 2735 . . . . . . 7 (0g𝑍) = (0g𝑍)
14 eqid 2735 . . . . . . 7 (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))
1512, 13, 14c0rhm 20551 . . . . . 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 2735 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
20 simpr 484 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21 zrtermoringc.e . . . . . . . . . . . . 13 (𝜑𝑍𝑈)
2210eldifad 3975 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ Ring)
2321, 22elind 4210 . . . . . . . . . . . 12 (𝜑𝑍 ∈ (𝑈 ∩ Ring))
2423, 4eleqtrrd 2842 . . . . . . . . . . 11 (𝜑𝑍 ∈ (Base‘𝐶))
2524adantr 480 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
261, 2, 18, 19, 20, 25ringchom 20669 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
2726eqcomd 2741 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
2827eleq2d 2825 . . . . . . 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 2825 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝐶)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ∈ (𝑟 RingHom 𝑍)))
31 eqid 2735 . . . . . . . . . . 11 (Base‘𝑍) = (Base‘𝑍)
3212, 31rhmf 20502 . . . . . . . . . 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 6737 . . . . . . . . . . 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 6920 . . . . . . . . . . . 12 (0g𝑍) ∈ V
3837, 14fnmpti 6712 . . . . . . . . . . 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 20545 . . . . . . . . . . . . . . . . 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 6724 . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)))
50 eqidd 2736 . . . . . . . . . . . . 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 2778 . . . . . . . . . 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 1925 . . . . . 6 (((𝜑𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀( ∈ (𝑟(Hom ‘𝐶)𝑍) → = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍))))
6017, 29, 593jca 1127 . . . . 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 2827 . . . . 5 ( = (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) → ( ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
6362eqeu 3715 . . . 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 3144 . 2 (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃! ∈ (𝑟(Hom ‘𝐶)𝑍))
661ringccat 20680 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
673, 66syl 17 . . 3 (𝜑𝐶 ∈ Cat)
682, 19, 67, 24istermo 18051 . 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 1086  wal 1535   = wceq 1537  wcel 2106  ∃!weu 2566  wral 3059  Vcvv 3478  cdif 3960  cin 3962  {csn 4631  cmpt 5231   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  0gc0g 17486  Catccat 17709  TermOctermo 18036  Ringcrg 20251   RingHom crh 20486  NzRingcnzr 20529  RingCatcringc 20662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-hom 17322  df-cco 17323  df-0g 17488  df-cat 17713  df-cid 17714  df-homf 17715  df-ssc 17858  df-resc 17859  df-subc 17860  df-termo 18039  df-estrc 18178  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-rhm 20489  df-nzr 20530  df-ringc 20663
This theorem is referenced by:  nzerooringczr  21509
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