Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubc | Structured version Visualization version GIF version |
Description: According to df-subc 17070, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17098 and subcss2 17101). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhm.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcrescrhm.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhm.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubc | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
3 | eqidd 2819 | . . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈)) | |
4 | 1, 2, 3 | rhmsscrnghm 44225 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
5 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))) |
7 | rngcrescrhm.c | . . . . . . 7 ⊢ 𝐶 = (RngCat‘𝑈) | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈)) |
9 | 8 | eqcomd 2824 | . . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶) |
10 | 9 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶)) |
11 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
12 | 7, 11, 1 | rngchomfeqhom 44168 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
13 | eqid 2818 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
14 | 7, 11, 1, 13 | rngchomfval 44165 | . . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
15 | 7, 11, 1 | rngcbas 44164 | . . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
16 | incom 4175 | . . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
17 | 15, 16 | syl6eq 2869 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈)) |
18 | 17 | sqxpeqd 5580 | . . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))) |
19 | 18 | reseq2d 5846 | . . . . 5 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
20 | 14, 19 | eqtrd 2853 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
21 | 10, 12, 20 | 3eqtrd 2857 | . . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
22 | 4, 6, 21 | 3brtr4d 5089 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈))) |
23 | 1, 7, 2, 5 | rhmsubclem3 44287 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
24 | 1, 7, 2, 5 | rhmsubclem4 44288 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
25 | 24 | ralrimivva 3188 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
26 | 25 | ralrimivva 3188 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
27 | 23, 26 | jca 512 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
28 | 27 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
29 | eqid 2818 | . . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈)) | |
30 | eqid 2818 | . . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈)) | |
31 | eqid 2818 | . . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈)) | |
32 | eqid 2818 | . . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈) | |
33 | 32 | rngccat 44177 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat) |
34 | 1, 33 | syl 17 | . . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat) |
35 | 1, 7, 2, 5 | rhmsubclem1 44285 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
36 | 29, 30, 31, 34, 35 | issubc2 17094 | . 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘(RngCat‘𝑈)) ↔ (𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)) ∧ ∀𝑥 ∈ 𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))))) |
37 | 22, 28, 36 | mpbir2and 709 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 〈cop 4563 class class class wbr 5057 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 compcco 16565 Catccat 16923 Idccid 16924 Homf chomf 16925 ⊆cat cssc 17065 Subcatcsubc 17067 Ringcrg 19226 RingHom crh 19393 Rngcrng 44073 RngHomo crngh 44084 RngCatcrngc 44156 |
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-rmo 3143 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-map 8397 df-pm 8398 df-ixp 8450 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-sets 16478 df-ress 16479 df-plusg 16566 df-hom 16577 df-cco 16578 df-0g 16703 df-cat 16927 df-cid 16928 df-homf 16929 df-ssc 17068 df-resc 17069 df-subc 17070 df-estrc 17361 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-rnghom 19396 df-mgmhm 43923 df-rng0 44074 df-rnghomo 44086 df-rngc 44158 |
This theorem is referenced by: rhmsubccat 44290 |
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