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Theorem rngcresringcat 41822
 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 2611 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4 eqidd 2611 . . . 4 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5 eqidd 2611 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
61, 2, 3, 4, 5dfrngc2 41764 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7 inex1g 4729 . . . 4 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
82, 7syl 17 . . 3 (𝜑 → (𝑈 ∩ Rng) ∈ V)
9 rnghmfn 41680 . . . . 5 RngHomo Fn (Rng × Rng)
10 fnfun 5902 . . . . 5 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
119, 10mp1i 13 . . . 4 (𝜑 → Fun RngHomo )
12 sqxpexg 6861 . . . . 5 ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
138, 12syl 17 . . . 4 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14 resfunexg 6384 . . . 4 ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
1511, 13, 14syl2anc 691 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16 fvex 6113 . . . 4 (comp‘(ExtStrCat‘𝑈)) ∈ V
1716a1i 11 . . 3 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
18 rhmsubcrngc.b . . . . 5 (𝜑𝐵 = (Ring ∩ 𝑈))
19 incom 3767 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
2018, 19syl6eq 2660 . . . 4 (𝜑𝐵 = (𝑈 ∩ Ring))
21 inex1g 4729 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
222, 21syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
2320, 22eqeltrd 2688 . . 3 (𝜑𝐵 ∈ V)
24 rhmsubcrngc.h . . . 4 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
25 rhmfn 41708 . . . . . 6 RingHom Fn (Ring × Ring)
26 fnfun 5902 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2725, 26mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
28 sqxpexg 6861 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2923, 28syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
30 resfunexg 6384 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3127, 29, 30syl2anc 691 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3224, 31eqeltrd 2688 . . 3 (𝜑𝐻 ∈ V)
33 ringrng 41669 . . . . . . 7 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
3433a1i 11 . . . . . 6 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3534ssrdv 3574 . . . . 5 (𝜑 → Ring ⊆ Rng)
36 ssrin 3800 . . . . 5 (Ring ⊆ Rng → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
3735, 36syl 17 . . . 4 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
38 incom 3767 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
3938a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
4037, 18, 393sstr4d 3611 . . 3 (𝜑𝐵 ⊆ (𝑈 ∩ Rng))
416, 8, 15, 17, 23, 32, 40estrres 16602 . 2 (𝜑 → ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
42 eqid 2610 . . 3 (𝐶cat 𝐻) = (𝐶cat 𝐻)
431a1i 11 . . . 4 (𝜑𝐶 = (RngCat‘𝑈))
44 fvex 6113 . . . . 5 (RngCat‘𝑈) ∈ V
4544a1i 11 . . . 4 (𝜑 → (RngCat‘𝑈) ∈ V)
4643, 45eqeltrd 2688 . . 3 (𝜑𝐶 ∈ V)
4720, 24rhmresfn 41801 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
4842, 46, 23, 47rescval2 16311 . 2 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
49 eqid 2610 . . 3 (RingCat‘𝑈) = (RingCat‘𝑈)
5049, 2, 20, 24, 5dfringc2 41810 . 2 (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
5141, 48, 503eqtr4d 2654 1 (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {ctp 4129  ⟨cop 4131   × cxp 5036   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾s cress 15696  Hom chom 15779  compcco 15780   ↾cat cresc 16291  ExtStrCatcestrc 16585  Ringcrg 18370   RingHom crh 18535  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749  RingCatcringc 41795 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-resc 16294  df-estrc 16586  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-ghm 17481  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-rnghom 18538  df-rng0 41665  df-rnghomo 41677  df-rngc 41751  df-ringc 41797 This theorem is referenced by: (None)
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