Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngchomrnghmresALTV Structured version   Visualization version   GIF version

Theorem rngchomrnghmresALTV 41788
 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 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 rngchomrnghmresALTV.u . . . . 5 (𝜑𝑈𝑉)
41, 2, 3rngcbasALTV 41775 . . . 4 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5 inss2 3796 . . . 4 (𝑈 ∩ Rng) ⊆ Rng
64, 5syl6eqss 3618 . . 3 (𝜑 → (Base‘𝐶) ⊆ Rng)
7 resmpt2 6656 . . 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 691 . 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 41677 . . . . . 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 6577 . . . . . . . . 9 (𝑤𝑚 𝑣) ∈ V
1110rabex 4740 . . . . . . . 8 {𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1211csbex 4721 . . . . . . 7 (Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1312csbex 4721 . . . . . 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 7128 . . . . 5 RngHomo Fn (Rng × Rng)
1514a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
16 fnov 6666 . . . 4 ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
1715, 16sylib 207 . . 3 (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
18 incom 3767 . . . . . 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 2655 . . . 4 (𝜑𝐵 = (Base‘𝐶))
2322sqxpeqd 5065 . . 3 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
2417, 23reseq12d 5318 . 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 41787 . 2 (𝜑𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
278, 24, 263eqtr4rd 2655 1 (𝜑𝐹 = ( RngHomo ↾ (𝐵 × 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ⦋csb 3499   ∩ cin 3539   ⊆ wss 3540   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Homf chomf 16150  Rngcrng 41664   RngHomo crngh 41675  RngCatALTVcrngcALTV 41750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-homf 16154  df-rnghomo 41677  df-rngcALTV 41752 This theorem is referenced by:  rhmsubcALTV  41901
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