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Theorem rngchomrnghmresALTV 47264
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 𝐢 = (RngCatALTVβ€˜π‘ˆ)
rngchomrnghmresALTV.b 𝐡 = (Rng ∩ π‘ˆ)
rngchomrnghmresALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngchomrnghmresALTV.f 𝐹 = (Homf β€˜πΆ)
Assertion
Ref Expression
rngchomrnghmresALTV (πœ‘ β†’ 𝐹 = ( RngHom β†Ύ (𝐡 Γ— 𝐡)))

Proof of Theorem rngchomrnghmresALTV
Dummy variables π‘₯ 𝑦 𝑠 π‘Ÿ 𝑣 𝑀 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngchomrnghmresALTV.c . . . . 5 𝐢 = (RngCatALTVβ€˜π‘ˆ)
2 eqid 2727 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
3 rngchomrnghmresALTV.u . . . . 5 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
41, 2, 3rngcbasALTV 47251 . . . 4 (πœ‘ β†’ (Baseβ€˜πΆ) = (π‘ˆ ∩ Rng))
5 inss2 4225 . . . 4 (π‘ˆ ∩ Rng) βŠ† Rng
64, 5eqsstrdi 4032 . . 3 (πœ‘ β†’ (Baseβ€˜πΆ) βŠ† Rng)
7 resmpo 7534 . . 3 (((Baseβ€˜πΆ) βŠ† Rng ∧ (Baseβ€˜πΆ) βŠ† Rng) β†’ ((π‘₯ ∈ Rng, 𝑦 ∈ Rng ↦ (π‘₯ RngHom 𝑦)) β†Ύ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))) = (π‘₯ ∈ (Baseβ€˜πΆ), 𝑦 ∈ (Baseβ€˜πΆ) ↦ (π‘₯ RngHom 𝑦)))
86, 6, 7syl2anc 583 . 2 (πœ‘ β†’ ((π‘₯ ∈ Rng, 𝑦 ∈ Rng ↦ (π‘₯ RngHom 𝑦)) β†Ύ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))) = (π‘₯ ∈ (Baseβ€˜πΆ), 𝑦 ∈ (Baseβ€˜πΆ) ↦ (π‘₯ RngHom 𝑦)))
9 df-rnghm 20364 . . . . . 6 RngHom = (π‘Ÿ ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Baseβ€˜π‘Ÿ) / π‘£β¦Œβ¦‹(Baseβ€˜π‘ ) / π‘€β¦Œ{𝑓 ∈ (𝑀 ↑m 𝑣) ∣ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦)))})
10 ovex 7447 . . . . . . . . 9 (𝑀 ↑m 𝑣) ∈ V
1110rabex 5328 . . . . . . . 8 {𝑓 ∈ (𝑀 ↑m 𝑣) ∣ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦)))} ∈ V
1211csbex 5305 . . . . . . 7 ⦋(Baseβ€˜π‘ ) / π‘€β¦Œ{𝑓 ∈ (𝑀 ↑m 𝑣) ∣ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦)))} ∈ V
1312csbex 5305 . . . . . 6 ⦋(Baseβ€˜π‘Ÿ) / π‘£β¦Œβ¦‹(Baseβ€˜π‘ ) / π‘€β¦Œ{𝑓 ∈ (𝑀 ↑m 𝑣) ∣ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦)))} ∈ V
149, 13fnmpoi 8068 . . . . 5 RngHom Fn (Rng Γ— Rng)
1514a1i 11 . . . 4 (πœ‘ β†’ RngHom Fn (Rng Γ— Rng))
16 fnov 7546 . . . 4 ( RngHom Fn (Rng Γ— Rng) ↔ RngHom = (π‘₯ ∈ Rng, 𝑦 ∈ Rng ↦ (π‘₯ RngHom 𝑦)))
1715, 16sylib 217 . . 3 (πœ‘ β†’ RngHom = (π‘₯ ∈ Rng, 𝑦 ∈ Rng ↦ (π‘₯ RngHom 𝑦)))
18 incom 4197 . . . . . 6 (π‘ˆ ∩ Rng) = (Rng ∩ π‘ˆ)
1918a1i 11 . . . . 5 (πœ‘ β†’ (π‘ˆ ∩ Rng) = (Rng ∩ π‘ˆ))
20 rngchomrnghmresALTV.b . . . . . 6 𝐡 = (Rng ∩ π‘ˆ)
2120a1i 11 . . . . 5 (πœ‘ β†’ 𝐡 = (Rng ∩ π‘ˆ))
2219, 4, 213eqtr4rd 2778 . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΆ))
2322sqxpeqd 5704 . . 3 (πœ‘ β†’ (𝐡 Γ— 𝐡) = ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))
2417, 23reseq12d 5980 . 2 (πœ‘ β†’ ( RngHom β†Ύ (𝐡 Γ— 𝐡)) = ((π‘₯ ∈ Rng, 𝑦 ∈ Rng ↦ (π‘₯ RngHom 𝑦)) β†Ύ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))))
25 rngchomrnghmresALTV.f . . 3 𝐹 = (Homf β€˜πΆ)
261, 2, 3, 25rngchomffvalALTV 47263 . 2 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ (Baseβ€˜πΆ), 𝑦 ∈ (Baseβ€˜πΆ) ↦ (π‘₯ RngHom 𝑦)))
278, 24, 263eqtr4rd 2778 1 (πœ‘ β†’ 𝐹 = ( RngHom β†Ύ (𝐡 Γ— 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  {crab 3427  β¦‹csb 3889   ∩ cin 3943   βŠ† wss 3944   Γ— cxp 5670   β†Ύ cres 5674   Fn wfn 6537  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑m cmap 8836  Basecbs 17171  +gcplusg 17224  .rcmulr 17225  Homf chomf 17637  Rngcrng 20083   RngHom crnghm 20362  RngCatALTVcrngcALTV 47248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-slot 17142  df-ndx 17154  df-base 17172  df-hom 17248  df-cco 17249  df-homf 17641  df-rnghm 20364  df-rngcALTV 47249
This theorem is referenced by:  rhmsubcALTV  47270
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