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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 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngchomrnghmresALTV.c | . . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
2 | eqid 2797 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | rngchomrnghmresALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | 1, 2, 3 | rngcbasALTV 43754 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
5 | inss2 4132 | . . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
6 | 4, 5 | syl6eqss 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))) | |
8 | 6, 6, 7 | syl2anc 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 ∈ (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 7055 | . . . . . . . . 9 ⊢ (𝑤 ↑𝑚 𝑣) ∈ V | |
11 | 10 | rabex 5133 | . . . . . . . 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 5113 | . . . . . . 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 5113 | . . . . . 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 7631 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → RngHomo Fn (Rng × Rng)) |
16 | fnov 7145 | . . . 4 ⊢ ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦))) | |
17 | 15, 16 | sylib 219 | . . 3 ⊢ (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦))) |
18 | incom 4105 | . . . . . 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 2844 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
23 | 22 | sqxpeqd 5482 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))) |
24 | 17, 23 | reseq12d 5742 | . 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = ((𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦)) ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
25 | rngchomrnghmresALTV.f | . . 3 ⊢ 𝐹 = (Homf ‘𝐶) | |
26 | 1, 2, 3, 25 | rngchomffvalALTV 43766 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦))) |
27 | 8, 24, 26 | 3eqtr4rd 2844 | 1 ⊢ (𝜑 → 𝐹 = ( 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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