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Theorem rngchomrnghmresALTV 43767
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 2797 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 rngchomrnghmresALTV.u . . . . 5 (𝜑𝑈𝑉)
41, 2, 3rngcbasALTV 43754 . . . 4 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5 inss2 4132 . . . 4 (𝑈 ∩ Rng) ⊆ Rng
64, 5syl6eqss 3948 . . 3 (𝜑 → (Base‘𝐶) ⊆ Rng)
7 resmpo 7135 . . 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 584 . 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 43658 . . . . . 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 7055 . . . . . . . . 9 (𝑤𝑚 𝑣) ∈ V
1110rabex 5133 . . . . . . . 8 {𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1211csbex 5113 . . . . . . 7 (Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} ∈ V
1312csbex 5113 . . . . . 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 7631 . . . . 5 RngHomo Fn (Rng × Rng)
1514a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
16 fnov 7145 . . . 4 ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
1715, 16sylib 219 . . 3 (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)))
18 incom 4105 . . . . . 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 2844 . . . 4 (𝜑𝐵 = (Base‘𝐶))
2322sqxpeqd 5482 . . 3 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
2417, 23reseq12d 5742 . 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 43766 . 2 (𝜑𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦)))
278, 24, 263eqtr4rd 2844 1 (𝜑𝐹 = ( RngHomo ↾ (𝐵 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  wral 3107  {crab 3111  csb 3817  cin 3864  wss 3865   × cxp 5448  cres 5452   Fn wfn 6227  cfv 6232  (class class class)co 7023  cmpo 7025  𝑚 cmap 8263  Basecbs 16316  +gcplusg 16398  .rcmulr 16399  Homf chomf 16770  Rngcrng 43645   RngHomo crngh 43656  RngCatALTVcrngcALTV 43729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-fz 12747  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-hom 16422  df-cco 16423  df-homf 16774  df-rnghomo 43658  df-rngcALTV 43731
This theorem is referenced by:  rhmsubcALTV  43879
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