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Theorem rngcresringcat 41821
Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)
Hypotheses
Ref Expression
rhmsubcrngc.c 𝐶 = (RngCat‘𝑈)
rhmsubcrngc.u (𝜑𝑈𝑉)
rhmsubcrngc.b (𝜑𝐵 = (Ring ∩ 𝑈))
rhmsubcrngc.h (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rngcresringcat (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))

Proof of Theorem rngcresringcat
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 rhmsubcrngc.c . . . 4 𝐶 = (RngCat‘𝑈)
2 rhmsubcrngc.u . . . 4 (𝜑𝑈𝑉)
3 eqidd 2607 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4 eqidd 2607 . . . 4 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5 eqidd 2607 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
61, 2, 3, 4, 5dfrngc2 41763 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7 inex1g 4721 . . . 4 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
82, 7syl 17 . . 3 (𝜑 → (𝑈 ∩ Rng) ∈ V)
9 rnghmfn 41679 . . . . 5 RngHomo Fn (Rng × Rng)
10 fnfun 5885 . . . . 5 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
119, 10mp1i 13 . . . 4 (𝜑 → Fun RngHomo )
12 sqxpexg 6835 . . . . 5 ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
138, 12syl 17 . . . 4 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14 resfunexg 6359 . . . 4 ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
1511, 13, 14syl2anc 690 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16 fvex 6095 . . . 4 (comp‘(ExtStrCat‘𝑈)) ∈ V
1716a1i 11 . . 3 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
18 rhmsubcrngc.b . . . . 5 (𝜑𝐵 = (Ring ∩ 𝑈))
19 incom 3763 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
2018, 19syl6eq 2656 . . . 4 (𝜑𝐵 = (𝑈 ∩ Ring))
21 inex1g 4721 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
222, 21syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
2320, 22eqeltrd 2684 . . 3 (𝜑𝐵 ∈ V)
24 rhmsubcrngc.h . . . 4 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
25 rhmfn 41707 . . . . . 6 RingHom Fn (Ring × Ring)
26 fnfun 5885 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2725, 26mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
28 sqxpexg 6835 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2923, 28syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
30 resfunexg 6359 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3127, 29, 30syl2anc 690 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3224, 31eqeltrd 2684 . . 3 (𝜑𝐻 ∈ V)
33 ringrng 41668 . . . . . . 7 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
3433a1i 11 . . . . . 6 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3534ssrdv 3570 . . . . 5 (𝜑 → Ring ⊆ Rng)
36 ssrin 3796 . . . . 5 (Ring ⊆ Rng → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
3735, 36syl 17 . . . 4 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
38 incom 3763 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
3938a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
4037, 18, 393sstr4d 3607 . . 3 (𝜑𝐵 ⊆ (𝑈 ∩ Rng))
416, 8, 15, 17, 23, 32, 40estrres 16545 . 2 (𝜑 → ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
42 eqid 2606 . . 3 (𝐶cat 𝐻) = (𝐶cat 𝐻)
431a1i 11 . . . 4 (𝜑𝐶 = (RngCat‘𝑈))
44 fvex 6095 . . . . 5 (RngCat‘𝑈) ∈ V
4544a1i 11 . . . 4 (𝜑 → (RngCat‘𝑈) ∈ V)
4643, 45eqeltrd 2684 . . 3 (𝜑𝐶 ∈ V)
4720, 24rhmresfn 41800 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
4842, 46, 23, 47rescval2 16254 . 2 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
49 eqid 2606 . . 3 (RingCat‘𝑈) = (RingCat‘𝑈)
5049, 2, 20, 24, 5dfringc2 41809 . 2 (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
5141, 48, 503eqtr4d 2650 1 (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3169  cin 3535  wss 3536  {ctp 4125  cop 4127   × cxp 5023  cres 5027  Fun wfun 5781   Fn wfn 5782  cfv 5787  (class class class)co 6524  ndxcnx 15635   sSet csts 15636  Basecbs 15638  s cress 15639  Hom chom 15722  compcco 15723  cat cresc 16234  ExtStrCatcestrc 16528  Ringcrg 18313   RingHom crh 18478  Rngcrng 41663   RngHomo crngh 41674  RngCatcrngc 41748  RingCatcringc 41794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-fz 12150  df-struct 15640  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-ress 15645  df-plusg 15724  df-hom 15736  df-cco 15737  df-0g 15868  df-resc 16237  df-estrc 16529  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-mhm 17101  df-grp 17191  df-minusg 17192  df-ghm 17424  df-cmn 17961  df-abl 17962  df-mgp 18256  df-ur 18268  df-ring 18315  df-rnghom 18481  df-rng0 41664  df-rnghomo 41676  df-rngc 41750  df-ringc 41796
This theorem is referenced by: (None)
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