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Theorem rngchomrnghmresALTV 44607
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 (𝜑𝑈𝑉)
rngchomrnghmresALTV.f 𝐹 = (Homf𝐶)
Assertion
Ref Expression
rngchomrnghmresALTV (𝜑𝐹 = ( 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.
StepHypRef Expression
1 rngchomrnghmresALTV.c . . . . 5 𝐶 = (RngCatALTV‘𝑈)
2 eqid 2801 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 rngchomrnghmresALTV.u . . . . 5 (𝜑𝑈𝑉)
41, 2, 3rngcbasALTV 44594 . . . 4 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5 inss2 4159 . . . 4 (𝑈 ∩ Rng) ⊆ Rng
64, 5eqsstrdi 3972 . . 3 (𝜑 → (Base‘𝐶) ⊆ Rng)
7 resmpo 7255 . . 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)))
86, 6, 7syl2anc 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 44498 . . . . . 6 RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
10 ovex 7172 . . . . . . . . 9 (𝑤m 𝑣) ∈ V
1110rabex 5202 . . . . . . . 8 {𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1211csbex 5182 . . . . . . 7 (Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1312csbex 5182 . . . . . 6 (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
149, 13fnmpoi 7754 . . . . 5 RngHomo Fn (Rng × Rng)
1514a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
16 fnov 7265 . . . 4 ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
1715, 16sylib 221 . . 3 (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
18 incom 4131 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1918a1i 11 . . . . 5 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
20 rngchomrnghmresALTV.b . . . . . 6 𝐵 = (Rng ∩ 𝑈)
2120a1i 11 . . . . 5 (𝜑𝐵 = (Rng ∩ 𝑈))
2219, 4, 213eqtr4rd 2847 . . . 4 (𝜑𝐵 = (Base‘𝐶))
2322sqxpeqd 5555 . . 3 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
2417, 23reseq12d 5823 . 2 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))))
25 rngchomrnghmresALTV.f . . 3 𝐹 = (Homf𝐶)
261, 2, 3, 25rngchomffvalALTV 44606 . 2 (𝜑𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
278, 24, 263eqtr4rd 2847 1 (𝜑𝐹 = ( RngHomo ↾ (𝐵 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  csb 3831  cin 3883  wss 3884   × cxp 5521  cres 5525   Fn wfn 6323  cfv 6328  (class class class)co 7139  cmpo 7141  m cmap 8393  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Homf chomf 16932  Rngcrng 44485   RngHomo crngh 44496  RngCatALTVcrngcALTV 44569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-homf 16936  df-rnghomo 44498  df-rngcALTV 44571
This theorem is referenced by:  rhmsubcALTV  44719
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