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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTV | Structured version Visualization version GIF version |
Description: According to df-subc 17074, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17102 and subcss2 17105). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhmALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcrescrhmALTV.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhmALTV.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubcALTV | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhmALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rngcrescrhmALTV.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
3 | eqidd 2799 | . . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈)) | |
4 | 1, 2, 3 | rhmsscrnghm 44650 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
5 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))) |
7 | eqid 2798 | . . . 4 ⊢ (RngCatALTV‘𝑈) = (RngCatALTV‘𝑈) | |
8 | eqid 2798 | . . . 4 ⊢ (Rng ∩ 𝑈) = (Rng ∩ 𝑈) | |
9 | eqid 2798 | . . . 4 ⊢ (Homf ‘(RngCatALTV‘𝑈)) = (Homf ‘(RngCatALTV‘𝑈)) | |
10 | 7, 8, 1, 9 | rngchomrnghmresALTV 44620 | . . 3 ⊢ (𝜑 → (Homf ‘(RngCatALTV‘𝑈)) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
11 | 4, 6, 10 | 3brtr4d 5062 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCatALTV‘𝑈))) |
12 | rngcrescrhmALTV.c | . . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
13 | 1, 12, 2, 5 | rhmsubcALTVlem3 44730 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
14 | 1, 12, 2, 5 | rhmsubcALTVlem4 44731 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
15 | 14 | ralrimivva 3156 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
16 | 15 | ralrimivva 3156 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
17 | 13, 16 | jca 515 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
18 | 17 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
19 | eqid 2798 | . . 3 ⊢ (Id‘(RngCatALTV‘𝑈)) = (Id‘(RngCatALTV‘𝑈)) | |
20 | eqid 2798 | . . 3 ⊢ (comp‘(RngCatALTV‘𝑈)) = (comp‘(RngCatALTV‘𝑈)) | |
21 | 7 | rngccatALTV 44614 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCatALTV‘𝑈) ∈ Cat) |
22 | 1, 21 | syl 17 | . . 3 ⊢ (𝜑 → (RngCatALTV‘𝑈) ∈ Cat) |
23 | 1, 12, 2, 5 | rhmsubcALTVlem1 44728 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
24 | 9, 19, 20, 22, 23 | issubc2 17098 | . 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)) ↔ (𝐻 ⊆cat (Homf ‘(RngCatALTV‘𝑈)) ∧ ∀𝑥 ∈ 𝑅 (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))))) |
25 | 11, 18, 24 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 〈cop 4531 class class class wbr 5030 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 compcco 16569 Catccat 16927 Idccid 16928 Homf chomf 16929 ⊆cat cssc 17069 Subcatcsubc 17071 Ringcrg 19290 RingHom crh 19460 Rngcrng 44498 RngHomo crngh 44509 RngCatALTVcrngcALTV 44582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-hom 16581 df-cco 16582 df-0g 16707 df-cat 16931 df-cid 16932 df-homf 16933 df-ssc 17072 df-subc 17074 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-ghm 18348 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-rnghom 19463 df-mgmhm 44399 df-rng0 44499 df-rnghomo 44511 df-rngcALTV 44584 |
This theorem is referenced by: rhmsubcALTVcat 44733 |
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