Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | rngchomrnghmresALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | 1, 2, 3 | rngcbasALTV 44274 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
5 | inss2 4206 | . . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
6 | 4, 5 | eqsstrdi 4021 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng) |
7 | resmpo 7272 | . . 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 586 | . 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 44178 | . . . . . 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 7189 | . . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V | |
11 | 10 | rabex 5235 | . . . . . . . 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 5215 | . . . . . . 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 5215 | . . . . . 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 7768 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → RngHomo Fn (Rng × Rng)) |
16 | fnov 7282 | . . . 4 ⊢ ( RngHomo Fn (Rng × Rng) ↔ RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦))) | |
17 | 15, 16 | sylib 220 | . . 3 ⊢ (𝜑 → RngHomo = (𝑥 ∈ Rng, 𝑦 ∈ Rng ↦ (𝑥 RngHomo 𝑦))) |
18 | incom 4178 | . . . . . 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 2867 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
23 | 22 | sqxpeqd 5587 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))) |
24 | 17, 23 | reseq12d 5854 | . 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 44286 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦))) |
27 | 8, 24, 26 | 3eqtr4rd 2867 | 1 ⊢ (𝜑 → 𝐹 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ⦋csb 3883 ∩ cin 3935 ⊆ wss 3936 × cxp 5553 ↾ cres 5557 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑m cmap 8406 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Homf chomf 16937 Rngcrng 44165 RngHomo crngh 44176 RngCatALTVcrngcALTV 44249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-homf 16941 df-rnghomo 44178 df-rngcALTV 44251 |
This theorem is referenced by: rhmsubcALTV 44399 |
Copyright terms: Public domain | W3C validator |