Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcrngc | Structured version Visualization version GIF version |
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-Mar-2020.) |
Ref | Expression |
---|---|
rhmsubcrngc.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rhmsubcrngc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rhmsubcrngc.b | ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) |
rhmsubcrngc.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rhmsubcrngc | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcrngc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rhmsubcrngc.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
3 | eqid 2734 | . . . . . . 7 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈) | |
4 | eqid 2734 | . . . . . . 7 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈)) | |
5 | 3, 4, 1 | rngcbas 45150 | . . . . . 6 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
6 | incom 4105 | . . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
7 | 5, 6 | eqtrdi 2790 | . . . . 5 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (Rng ∩ 𝑈)) |
8 | 1, 2, 7 | rhmsscrnghm 45211 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
9 | rhmsubcrngc.c | . . . . . . . 8 ⊢ 𝐶 = (RngCat‘𝑈) | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈)) |
11 | 10 | fveq2d 6710 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(RngCat‘𝑈))) |
12 | 11 | sqxpeqd 5572 | . . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))) |
13 | 12 | reseq2d 5840 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
14 | 8, 13 | breqtrrd 5071 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
15 | rhmsubcrngc.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
16 | eqid 2734 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
17 | 9, 16, 1 | rngchomfeqhom 45154 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
18 | eqid 2734 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
19 | 9, 16, 1, 18 | rngchomfval 45151 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
20 | 17, 19 | eqtrd 2774 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
21 | 14, 15, 20 | 3brtr4d 5075 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶)) |
22 | 9, 1, 2, 15 | rhmsubcrngclem1 45212 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
23 | 9, 1, 2, 15 | rhmsubcrngclem2 45213 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
24 | 22, 23 | jca 515 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
25 | 24 | ralrimiva 3098 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
26 | eqid 2734 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
27 | eqid 2734 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
28 | eqid 2734 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
29 | 9 | rngccat 45163 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
30 | 1, 29 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
31 | incom 4105 | . . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
32 | 2, 31 | eqtrdi 2790 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
33 | 32, 15 | rhmresfn 45194 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
34 | 26, 27, 28, 30, 33 | issubc2 17314 | . 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘𝐶) ↔ (𝐻 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))))) |
35 | 21, 25, 34 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∩ cin 3856 〈cop 4537 class class class wbr 5043 × cxp 5538 ↾ cres 5542 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 Hom chom 16778 compcco 16779 Catccat 17139 Idccid 17140 Homf chomf 17141 ⊆cat cssc 17284 Subcatcsubc 17286 Ringcrg 19534 RingHom crh 19704 Rngcrng 45059 RngHomo crngh 45070 RngCatcrngc 45142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-hom 16791 df-cco 16792 df-0g 16918 df-cat 17143 df-cid 17144 df-homf 17145 df-ssc 17287 df-resc 17288 df-subc 17289 df-estrc 17602 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-grp 18340 df-minusg 18341 df-ghm 18592 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-rnghom 19707 df-mgmhm 44960 df-rng0 45060 df-rnghomo 45072 df-rngc 45144 df-ringc 45190 |
This theorem is referenced by: (None) |
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