Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngcresringcat Structured version   Visualization version   GIF version

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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