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Theorem rngcresringcat 41795
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 2621 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4 eqidd 2621 . . . 4 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5 eqidd 2621 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
61, 2, 3, 4, 5dfrngc2 41737 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7 inex1g 4792 . . . 4 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
82, 7syl 17 . . 3 (𝜑 → (𝑈 ∩ Rng) ∈ V)
9 rnghmfn 41655 . . . . 5 RngHomo Fn (Rng × Rng)
10 fnfun 5976 . . . . 5 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
119, 10mp1i 13 . . . 4 (𝜑 → Fun RngHomo )
12 sqxpexg 6948 . . . . 5 ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
138, 12syl 17 . . . 4 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14 resfunexg 6464 . . . 4 ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
1511, 13, 14syl2anc 692 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16 fvexd 6190 . . 3 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17 rhmsubcrngc.b . . . . 5 (𝜑𝐵 = (Ring ∩ 𝑈))
18 incom 3797 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
1917, 18syl6eq 2670 . . . 4 (𝜑𝐵 = (𝑈 ∩ Ring))
20 inex1g 4792 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
212, 20syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
2219, 21eqeltrd 2699 . . 3 (𝜑𝐵 ∈ V)
23 rhmsubcrngc.h . . . 4 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
24 rhmfn 41683 . . . . . 6 RingHom Fn (Ring × Ring)
25 fnfun 5976 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2624, 25mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
27 sqxpexg 6948 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2822, 27syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
29 resfunexg 6464 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3026, 28, 29syl2anc 692 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3123, 30eqeltrd 2699 . . 3 (𝜑𝐻 ∈ V)
32 ringrng 41644 . . . . . . 7 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
3332a1i 11 . . . . . 6 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3433ssrdv 3601 . . . . 5 (𝜑 → Ring ⊆ Rng)
35 ssrin 3830 . . . . 5 (Ring ⊆ Rng → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
3634, 35syl 17 . . . 4 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
37 incom 3797 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
3837a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
3936, 17, 383sstr4d 3640 . . 3 (𝜑𝐵 ⊆ (𝑈 ∩ Rng))
406, 8, 15, 16, 22, 31, 39estrres 16760 . 2 (𝜑 → ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
41 eqid 2620 . . 3 (𝐶cat 𝐻) = (𝐶cat 𝐻)
421a1i 11 . . . 4 (𝜑𝐶 = (RngCat‘𝑈))
43 fvexd 6190 . . . 4 (𝜑 → (RngCat‘𝑈) ∈ V)
4442, 43eqeltrd 2699 . . 3 (𝜑𝐶 ∈ V)
4519, 23rhmresfn 41774 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
4641, 44, 22, 45rescval2 16469 . 2 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
47 eqid 2620 . . 3 (RingCat‘𝑈) = (RingCat‘𝑈)
4847, 2, 19, 23, 5dfringc2 41783 . 2 (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
4940, 46, 483eqtr4d 2664 1 (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  wss 3567  {ctp 4172  cop 4174   × cxp 5102  cres 5106  Fun wfun 5870   Fn wfn 5871  cfv 5876  (class class class)co 6635  ndxcnx 15835   sSet csts 15836  Basecbs 15838  s cress 15839  Hom chom 15933  compcco 15934  cat cresc 16449  ExtStrCatcestrc 16743  Ringcrg 18528   RingHom crh 18693  Rngcrng 41639   RngHomo crngh 41650  RngCatcrngc 41722  RingCatcringc 41768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-hom 15947  df-cco 15948  df-0g 16083  df-resc 16452  df-estrc 16744  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-mhm 17316  df-grp 17406  df-minusg 17407  df-ghm 17639  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-rnghom 18696  df-rng0 41640  df-rnghomo 41652  df-rngc 41724  df-ringc 41770
This theorem is referenced by: (None)
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