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 2818 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | rngchomrnghmresALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | 1, 2, 3 | rngcbasALTV 44182 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
5 | inss2 4203 | . . . 4 ⊢ (𝑈 ∩ Rng) ⊆ Rng | |
6 | 4, 5 | eqsstrdi 4018 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ Rng) |
7 | resmpo 7261 | . . 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 44086 | . . . . . 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 7178 | . . . . . . . . 9 ⊢ (𝑤 ↑m 𝑣) ∈ V | |
11 | 10 | rabex 5226 | . . . . . . . 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 5206 | . . . . . . 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 5206 | . . . . . 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 7757 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → RngHomo Fn (Rng × Rng)) |
16 | fnov 7271 | . . . 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 4175 | . . . . . 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 2864 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
23 | 22 | sqxpeqd 5580 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))) |
24 | 17, 23 | reseq12d 5847 | . 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 44194 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥 RngHomo 𝑦))) |
27 | 8, 24, 26 | 3eqtr4rd 2864 | 1 ⊢ (𝜑 → 𝐹 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ⦋csb 3880 ∩ cin 3932 ⊆ wss 3933 × cxp 5546 ↾ cres 5550 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Homf chomf 16925 Rngcrng 44073 RngHomo crngh 44084 RngCatALTVcrngcALTV 44157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-homf 16929 df-rnghomo 44086 df-rngcALTV 44159 |
This theorem is referenced by: rhmsubcALTV 44307 |
Copyright terms: Public domain | W3C validator |