Theorem rngchomrnghmresALTV 48273

Description: The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rngchomrnghmresALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngchomrnghmresALTV.b	⊢ 𝐵 = (Rng ∩ 𝑈)
rngchomrnghmresALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngchomrnghmresALTV.f	⊢ 𝐹 = (Homf ‘𝐶)

Assertion

Ref	Expression
rngchomrnghmresALTV	⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))

Proof of Theorem rngchomrnghmresALTV

Dummy variables 𝑥 𝑦 𝑠 𝑟 𝑣 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngchomrnghmresALTV.c	. . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈)
2		eqid 2729	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		rngchomrnghmresALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbasALTV 48260	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5		inss2 4189	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng
6	4, 5	eqsstrdi 3980	. . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng)
7		resmpo 7469	. . 3 ⊢ (((Base‘𝐶) ⊆ Rng ∧ (Base‘𝐶) ⊆ Rng) → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
8	6, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
9		df-rnghm 20321	. . . . . 6 ⊢ RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))})
10		ovex 7382	. . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V
11	10	rabex 5278	. . . . . . . 8 ⊢ {𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V
12	11	csbex 5250	. . . . . . 7 ⊢ ⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V
13	12	csbex 5250	. . . . . 6 ⊢ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V
14	9, 13	fnmpoi 8005	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
16		fnov 7480	. . . 4 ⊢ ( RngHom Fn (Rng × Rng) ↔ RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)))
17	15, 16	sylib 218	. . 3 ⊢ (𝜑 → RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)))
18		incom 4160	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
20		rngchomrnghmresALTV.b	. . . . . 6 ⊢ 𝐵 = (Rng ∩ 𝑈)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
22	19, 4, 21	3eqtr4rd 2775	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
23	22	sqxpeqd 5651	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
24	17, 23	reseq12d 5931	. 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))))
25		rngchomrnghmresALTV.f	. . 3 ⊢ 𝐹 = (Homf ‘𝐶)
26	1, 2, 3, 25	rngchomffvalALTV 48272	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
27	8, 24, 26	3eqtr4rd 2775	1 ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  ⦋csb 3851   ∩ cin 3902   ⊆ wss 3903   × cxp 5617   ↾ cres 5621   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Hom_f chomf 17572  Rngcrng 20037   RngHom crnghm 20319  RngCatALTVcrngcALTV 48257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-homf 17576  df-rnghm 20321  df-rngcALTV 48258

This theorem is referenced by:  rhmsubcALTV  48279