Theorem rngcresringcat 20705

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rhmsubcrngc.b	⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))
rhmsubcrngc.h	⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rngcresringcat	⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Proof of Theorem rngcresringcat

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmsubcrngc.c	. . . 4 ⊢ 𝐶 = (RngCat‘𝑈)
2		rhmsubcrngc.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		eqidd 2762	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
4		eqidd 2762	. . . 4 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
5		eqidd 2762	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)))
6	1, 2, 3, 4, 5	dfrngc2 20664	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), (𝑈 ∩ Rng)⟩, ⟨(Hom ‘ndx), ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
7		inex1g 5272	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
9		rnghmfn 20474	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
10		fnfun 6615	. . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom )
11	9, 10	mp1i 13	. . . 4 ⊢ (𝜑 → Fun RngHom )
12		sqxpexg 7732	. . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V)
14		resfunexg 7193	. . . 4 ⊢ ((Fun RngHom ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
15	11, 13, 14	syl2anc 593	. . 3 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V)
16		fvexd 6876	. . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
17		rhmsubcrngc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
18		rhmfn 20534	. . . . . 6 ⊢ RingHom Fn (Ring × Ring)
19		fnfun 6615	. . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom )
20	18, 19	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun RingHom )
21		rhmsubcrngc.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))
22		incom 4159	. . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
23	21, 22	eqtrdi 2812	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
24		inex1g 5272	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
25	2, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V)
26	23, 25	eqeltrd 2861	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
27		sqxpexg 7732	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7193	. . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
30	20, 28, 29	syl2anc 593	. . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
31	17, 30	eqeltrd 2861	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		ringrng 20321	. . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
34	33	ssrdv 3940	. . . . 5 ⊢ (𝜑 → Ring ⊆ Rng)
35	34	ssrind 4193	. . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
36		incom 4159	. . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
38	35, 21, 37	3sstr4d 3989	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng))
39	6, 8, 15, 16, 31, 38	estrres 18161	. 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
40		eqid 2761	. . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
41		fvexd 6876	. . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V)
42	1, 41	eqeltrid 2865	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
43	23, 17	rhmresfn 20684	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
44	40, 42, 26, 43	rescval2 17851	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
45		eqid 2761	. . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈)
46	45, 2, 23, 17, 5	dfringc2 20693	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), (comp‘(ExtStrCat‘𝑈))⟩})
47	39, 44, 46	3eqtr4d 2806	1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3901  {ctp 4583  ⟨cop 4585   × cxp 5641   ↾ cres 5645  Fun wfun 6509   Fn wfn 6510  ‘cfv 6515  (class class class)co 7390   sSet csts 17189  ndxcnx 17219  Basecbs 17235   ↾_s cress 17256  Hom chom 17287  compcco 17288   ↾_cat cresc 17831  ExtStrCatcestrc 18144  Rngcrng 20188  Ringcrg 20269   RngHom crnghm 20469   RingHom crh 20504  RngCatcrngc 20652  RingCatcringc 20681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-nn 12204  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12475  df-z 12562  df-dec 12682  df-uz 12833  df-fz 13506  df-struct 17173  df-sets 17190  df-slot 17208  df-ndx 17220  df-base 17236  df-ress 17257  df-plusg 17289  df-hom 17300  df-cco 17301  df-0g 17460  df-resc 17834  df-estrc 18145  df-mgm 18664  df-sgrp 18743  df-mnd 18759  df-mhm 18807  df-grp 18968  df-minusg 18969  df-ghm 19244  df-cmn 19812  df-abl 19813  df-mgp 20177  df-rng 20189  df-ur 20218  df-ring 20271  df-rnghm 20471  df-rhm 20507  df-rngc 20653  df-ringc 20682

This theorem is referenced by: (None)