Theorem rngchomrnghmresALTV 45181

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 = ( RngHomo ↾ (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 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		rngchomrnghmresALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbasALTV 45168	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5		inss2 4134	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng
6	4, 5	eqsstrdi 3945	. . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng)
7		resmpo 7319	. . 3 ⊢ (((Base‘𝐶) ⊆ Rng ∧ (Base‘𝐶) ⊆ Rng) → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
8	6, 6, 7	syl2anc 587	. 2 ⊢ (𝜑 → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
9		df-rnghomo 45072	. . . . . 6 ⊢ RngHomo = (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 7235	. . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V
11	10	rabex 5214	. . . . . . . 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 5193	. . . . . . 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 5193	. . . . . 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 7829	. . . . 5 ⊢ RngHomo Fn (Rng × Rng)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → RngHomo Fn (Rng × Rng))
16		fnov 7330	. . . 4 ⊢ ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
17	15, 16	sylib 221	. . 3 ⊢ (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
18		incom 4105	. . . . . 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 2785	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
23	22	sqxpeqd 5572	. . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
24	17, 23	reseq12d 5841	. 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))))
25		rngchomrnghmresALTV.f	. . 3 ⊢ 𝐹 = (Homf ‘𝐶)
26	1, 2, 3, 25	rngchomffvalALTV 45180	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
27	8, 24, 26	3eqtr4rd 2785	1 ⊢ (𝜑 → 𝐹 = ( RngHomo ↾ (𝐵 × 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3054  {crab 3058  ⦋csb 3802   ∩ cin 3856   ⊆ wss 3857   × cxp 5538   ↾ cres 5542   Fn wfn 6364  ‘cfv 6369  (class class class)co 7202   ∈ cmpo 7204   ↑_m cmap 8497  Basecbs 16684  +_gcplusg 16767  ._rcmulr 16768  Hom_f chomf 17141  Rngcrng 45059   RngHomo crngh 45070  RngCatALTVcrngcALTV 45143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-fz 13079  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-hom 16791  df-cco 16792  df-homf 17145  df-rnghomo 45072  df-rngcALTV 45145

This theorem is referenced by:  rhmsubcALTV  45293