Theorem rngchomrnghmresALTV 48200

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 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		rngchomrnghmresALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbasALTV 48187	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5		inss2 4237	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng
6	4, 5	eqsstrdi 4027	. . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng)
7		resmpo 7554	. . 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 20437	. . . . . 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 7465	. . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V
11	10	rabex 5338	. . . . . . . 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 5310	. . . . . . 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 5310	. . . . . 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 8096	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
16		fnov 7565	. . . 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 4208	. . . . . 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 2787	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
23	22	sqxpeqd 5716	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
24	17, 23	reseq12d 5997	. 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 48199	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
27	8, 24, 26	3eqtr4rd 2787	1 ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  {crab 3435  ⦋csb 3898   ∩ cin 3949   ⊆ wss 3950   × cxp 5682   ↾ cres 5686   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434   ↑_m cmap 8867  Basecbs 17248  +_gcplusg 17298  ._rcmulr 17299  Hom_f chomf 17710  Rngcrng 20150   RngHom crnghm 20435  RngCatALTVcrngcALTV 48184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-homf 17714  df-rnghm 20437  df-rngcALTV 48185

This theorem is referenced by:  rhmsubcALTV  48206