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Theorem rngchomrnghmresALTV 42475
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 2748 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 rngchomrnghmresALTV.u . . . . 5 (𝜑𝑈𝑉)
41, 2, 3rngcbasALTV 42462 . . . 4 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5 inss2 3965 . . . 4 (𝑈 ∩ Rng) ⊆ Rng
64, 5syl6eqss 3784 . . 3 (𝜑 → (Base‘𝐶) ⊆ Rng)
7 resmpt2 6911 . . 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 696 . 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 42366 . . . . . 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 6829 . . . . . . . . 9 (𝑤𝑚 𝑣) ∈ V
1110rabex 4952 . . . . . . . 8 {𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1211csbex 4933 . . . . . . 7 (Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1312csbex 4933 . . . . . 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, 13fnmpt2i 7395 . . . . 5 RngHomo Fn (Rng × Rng)
1514a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
16 fnov 6921 . . . 4 ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
1715, 16sylib 208 . . 3 (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
18 incom 3936 . . . . . 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 2793 . . . 4 (𝜑𝐵 = (Base‘𝐶))
2322sqxpeqd 5286 . . 3 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
2417, 23reseq12d 5540 . 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 42474 . 2 (𝜑𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
278, 24, 263eqtr4rd 2793 1 (𝜑𝐹 = ( RngHomo ↾ (𝐵 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  wral 3038  {crab 3042  csb 3662  cin 3702  wss 3703   × cxp 5252  cres 5256   Fn wfn 6032  cfv 6037  (class class class)co 6801  cmpt2 6803  𝑚 cmap 8011  Basecbs 16030  +gcplusg 16114  .rcmulr 16115  Homf chomf 16499  Rngcrng 42353   RngHomo crngh 42364  RngCatALTVcrngcALTV 42437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-2 11242  df-3 11243  df-4 11244  df-5 11245  df-6 11246  df-7 11247  df-8 11248  df-9 11249  df-n0 11456  df-z 11541  df-dec 11657  df-uz 11851  df-fz 12491  df-struct 16032  df-ndx 16033  df-slot 16034  df-base 16036  df-hom 16139  df-cco 16140  df-homf 16503  df-rnghomo 42366  df-rngcALTV 42439
This theorem is referenced by:  rhmsubcALTV  42587
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