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Theorem rngcresringcat 43666
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 2779 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4 eqidd 2779 . . . 4 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5 eqidd 2779 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
61, 2, 3, 4, 5dfrngc2 43608 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7 inex1g 5081 . . . 4 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
82, 7syl 17 . . 3 (𝜑 → (𝑈 ∩ Rng) ∈ V)
9 rnghmfn 43526 . . . . 5 RngHomo Fn (Rng × Rng)
10 fnfun 6288 . . . . 5 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
119, 10mp1i 13 . . . 4 (𝜑 → Fun RngHomo )
12 sqxpexg 7296 . . . . 5 ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
138, 12syl 17 . . . 4 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14 resfunexg 6806 . . . 4 ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
1511, 13, 14syl2anc 576 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16 fvexd 6516 . . 3 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17 rhmsubcrngc.h . . . 4 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18 rhmfn 43554 . . . . . 6 RingHom Fn (Ring × Ring)
19 fnfun 6288 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2018, 19mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
21 rhmsubcrngc.b . . . . . . . 8 (𝜑𝐵 = (Ring ∩ 𝑈))
22 incom 4068 . . . . . . . 8 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
2321, 22syl6eq 2830 . . . . . . 7 (𝜑𝐵 = (𝑈 ∩ Ring))
24 inex1g 5081 . . . . . . . 8 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
252, 24syl 17 . . . . . . 7 (𝜑 → (𝑈 ∩ Ring) ∈ V)
2623, 25eqeltrd 2866 . . . . . 6 (𝜑𝐵 ∈ V)
27 sqxpexg 7296 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2826, 27syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
29 resfunexg 6806 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3020, 28, 29syl2anc 576 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3117, 30eqeltrd 2866 . . 3 (𝜑𝐻 ∈ V)
32 ringrng 43515 . . . . . . 7 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
3332a1i 11 . . . . . 6 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3433ssrdv 3866 . . . . 5 (𝜑 → Ring ⊆ Rng)
3534ssrind 4101 . . . 4 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36 incom 4068 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
3736a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
3835, 21, 373sstr4d 3906 . . 3 (𝜑𝐵 ⊆ (𝑈 ∩ Rng))
396, 8, 15, 16, 31, 38estrres 17250 . 2 (𝜑 → ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40 eqid 2778 . . 3 (𝐶cat 𝐻) = (𝐶cat 𝐻)
41 fvexd 6516 . . . 4 (𝜑 → (RngCat‘𝑈) ∈ V)
421, 41syl5eqel 2870 . . 3 (𝜑𝐶 ∈ V)
4323, 17rhmresfn 43645 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
4440, 42, 26, 43rescval2 16959 . 2 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45 eqid 2778 . . 3 (RingCat‘𝑈) = (RingCat‘𝑈)
4645, 2, 23, 17, 5dfringc2 43654 . 2 (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
4739, 44, 463eqtr4d 2824 1 (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  Vcvv 3415  cin 3830  {ctp 4446  cop 4448   × cxp 5406  cres 5410  Fun wfun 6184   Fn wfn 6185  cfv 6190  (class class class)co 6978  ndxcnx 16339   sSet csts 16340  Basecbs 16342  s cress 16343  Hom chom 16435  compcco 16436  cat cresc 16939  ExtStrCatcestrc 17233  Ringcrg 19023   RingHom crh 19190  Rngcrng 43510   RngHomo crngh 43521  RngCatcrngc 43593  RingCatcringc 43639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-map 8210  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-2 11506  df-3 11507  df-4 11508  df-5 11509  df-6 11510  df-7 11511  df-8 11512  df-9 11513  df-n0 11711  df-z 11797  df-dec 11915  df-uz 12062  df-fz 12712  df-struct 16344  df-ndx 16345  df-slot 16346  df-base 16348  df-sets 16349  df-ress 16350  df-plusg 16437  df-hom 16448  df-cco 16449  df-0g 16574  df-resc 16942  df-estrc 17234  df-mgm 17713  df-sgrp 17755  df-mnd 17766  df-mhm 17806  df-grp 17897  df-minusg 17898  df-ghm 18130  df-cmn 18671  df-abl 18672  df-mgp 18966  df-ur 18978  df-ring 19025  df-rnghom 19193  df-rng0 43511  df-rnghomo 43523  df-rngc 43595  df-ringc 43641
This theorem is referenced by: (None)
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