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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchomrnghmresALTV | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| rngchomrnghmresALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) | 
| rngchomrnghmresALTV.b | ⊢ 𝐵 = (Rng ∩ 𝑈) | 
| rngchomrnghmresALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) | 
| rngchomrnghmresALTV.f | ⊢ 𝐹 = (Homf ‘𝐶) | 
| Ref | Expression | 
|---|---|
| rngchomrnghmresALTV | ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rngchomrnghmresALTV.c | . . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 3 | rngchomrnghmresALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 4 | 1, 2, 3 | rngcbasALTV 48187 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) | 
| 5 | inss2 4237 | . . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
| 6 | 4, 5 | eqsstrdi 4027 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng) | 
| 7 | resmpo 7554 | . . 3 ⊢ (((Base‘𝐶) ⊆ Rng ∧ (Base‘𝐶) ⊆ Rng) → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦))) | |
| 8 | 6, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦))) | 
| 9 | df-rnghm 20437 | . . . . . 6 ⊢ RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))}) | |
| 10 | ovex 7465 | . . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V | |
| 11 | 10 | rabex 5338 | . . . . . . . 8 ⊢ {𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V | 
| 12 | 11 | csbex 5310 | . . . . . . 7 ⊢ ⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V | 
| 13 | 12 | csbex 5310 | . . . . . 6 ⊢ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))} ∈ V | 
| 14 | 9, 13 | fnmpoi 8096 | . . . . 5 ⊢ RngHom Fn (Rng × Rng) | 
| 15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng)) | 
| 16 | fnov 7565 | . . . 4 ⊢ ( RngHom Fn (Rng × Rng) ↔ RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦))) | |
| 17 | 15, 16 | sylib 218 | . . 3 ⊢ (𝜑 → RngHom = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦))) | 
| 18 | incom 4208 | . . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
| 19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) | 
| 20 | rngchomrnghmresALTV.b | . . . . . 6 ⊢ 𝐵 = (Rng ∩ 𝑈) | |
| 21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) | 
| 22 | 19, 4, 21 | 3eqtr4rd 2787 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | 
| 23 | 22 | sqxpeqd 5716 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))) | 
| 24 | 17, 23 | reseq12d 5997 | . 2 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHom 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶)))) | 
| 25 | rngchomrnghmresALTV.f | . . 3 ⊢ 𝐹 = (Homf ‘𝐶) | |
| 26 | 1, 2, 3, 25 | rngchomffvalALTV 48199 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHom 𝑦))) | 
| 27 | 8, 24, 26 | 3eqtr4rd 2787 | 1 ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 ⦋csb 3898 ∩ cin 3949 ⊆ wss 3950 × cxp 5682 ↾ cres 5686 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ↑m cmap 8867 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Homf chomf 17710 Rngcrng 20150 RngHom crnghm 20435 RngCatALTVcrngcALTV 48184 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-hom 17322 df-cco 17323 df-homf 17714 df-rnghm 20437 df-rngcALTV 48185 | 
| This theorem is referenced by: rhmsubcALTV 48206 | 
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