Theorem rngchomrnghmresALTV 48123

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 2735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		rngchomrnghmresALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbasALTV 48110	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5		inss2 4246	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng
6	4, 5	eqsstrdi 4050	. . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng)
7		resmpo 7553	. . 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 20453	. . . . . 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 7464	. . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V
11	10	rabex 5345	. . . . . . . 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 5317	. . . . . . 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 5317	. . . . . 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 8094	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
16		fnov 7564	. . . 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 4217	. . . . . 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 2786	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
23	22	sqxpeqd 5721	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
24	17, 23	reseq12d 6001	. 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 48122	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦)))
27	8, 24, 26	3eqtr4rd 2786	1 ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  ⦋csb 3908   ∩ cin 3962   ⊆ wss 3963   × cxp 5687   ↾ cres 5691   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8865  Basecbs 17245  +_gcplusg 17298  ._rcmulr 17299  Hom_f chomf 17711  Rngcrng 20170   RngHom crnghm 20451  RngCatALTVcrngcALTV 48107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-homf 17715  df-rnghm 20453  df-rngcALTV 48108

This theorem is referenced by:  rhmsubcALTV  48129