Theorem rngchomrnghmresALTV 48906

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 2764	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		rngchomrnghmresALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbasALTV 48893	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5		inss2 4191	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng
6	4, 5	eqsstrdi 3982	. . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng)
7		resmpo 7518	. . 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 593	. 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 20487	. . . . . 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 7431	. . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V
11	10	rabex 5297	. . . . . . . 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 5263	. . . . . . 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 5263	. . . . . 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 8053	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
16		fnov 7529	. . . 4 ⊢ ( RngHom Fn (Rng × Rng) ↔ RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)))
17	15, 16	sylib 220	. . 3 ⊢ (𝜑 → RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)))
18		incom 4163	. . . . . 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 2810	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
23	22	sqxpeqd 5681	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
24	17, 23	reseq12d 5968	. 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 48905	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
27	8, 24, 26	3eqtr4rd 2810	1 ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  ⦋csb 3854   ∩ cin 3905   ⊆ wss 3906   × cxp 5647   ↾ cres 5651   Fn wfn 6518  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400   ↑_m cmap 8810  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  Hom_f chomf 17700  Rngcrng 20200   RngHom crnghm 20485  RngCatALTVcrngcALTV 48890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-hom 17312  df-cco 17313  df-homf 17704  df-rnghm 20487  df-rngcALTV 48891

This theorem is referenced by:  rhmsubcALTV  48912