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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 17703, the subcategories (SubcatβπΆ) of a category πΆ are subsets of the homomorphisms of πΆ (see subcssc 17734 and subcss2 17737). 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 | β’ (π β π β π) |
rngcrescrhmALTV.c | β’ πΆ = (RngCatALTVβπ) |
rngcrescrhmALTV.r | β’ (π β π = (Ring β© π)) |
rngcrescrhmALTV.h | β’ π» = ( RingHom βΎ (π Γ π )) |
Ref | Expression |
---|---|
rhmsubcALTV | β’ (π β π» β (Subcatβ(RngCatALTVβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhmALTV.u | . . . 4 β’ (π β π β π) | |
2 | rngcrescrhmALTV.r | . . . 4 β’ (π β π = (Ring β© π)) | |
3 | eqidd 2734 | . . . 4 β’ (π β (Rng β© π) = (Rng β© π)) | |
4 | 1, 2, 3 | rhmsscrnghm 46414 | . . 3 β’ (π β ( RingHom βΎ (π Γ π )) βcat ( RngHomo βΎ ((Rng β© π) Γ (Rng β© π)))) |
5 | rngcrescrhmALTV.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
6 | 5 | a1i 11 | . . 3 β’ (π β π» = ( RingHom βΎ (π Γ π ))) |
7 | eqid 2733 | . . . 4 β’ (RngCatALTVβπ) = (RngCatALTVβπ) | |
8 | eqid 2733 | . . . 4 β’ (Rng β© π) = (Rng β© π) | |
9 | eqid 2733 | . . . 4 β’ (Homf β(RngCatALTVβπ)) = (Homf β(RngCatALTVβπ)) | |
10 | 7, 8, 1, 9 | rngchomrnghmresALTV 46384 | . . 3 β’ (π β (Homf β(RngCatALTVβπ)) = ( RngHomo βΎ ((Rng β© π) Γ (Rng β© π)))) |
11 | 4, 6, 10 | 3brtr4d 5141 | . 2 β’ (π β π» βcat (Homf β(RngCatALTVβπ))) |
12 | rngcrescrhmALTV.c | . . . . 5 β’ πΆ = (RngCatALTVβπ) | |
13 | 1, 12, 2, 5 | rhmsubcALTVlem3 46494 | . . . 4 β’ ((π β§ π₯ β π ) β ((Idβ(RngCatALTVβπ))βπ₯) β (π₯π»π₯)) |
14 | 1, 12, 2, 5 | rhmsubcALTVlem4 46495 | . . . . . 6 β’ ((((π β§ π₯ β π ) β§ (π¦ β π β§ π§ β π )) β§ (π β (π₯π»π¦) β§ π β (π¦π»π§))) β (π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§)) |
15 | 14 | ralrimivva 3194 | . . . . 5 β’ (((π β§ π₯ β π ) β§ (π¦ β π β§ π§ β π )) β βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§)) |
16 | 15 | ralrimivva 3194 | . . . 4 β’ ((π β§ π₯ β π ) β βπ¦ β π βπ§ β π βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§)) |
17 | 13, 16 | jca 513 | . . 3 β’ ((π β§ π₯ β π ) β (((Idβ(RngCatALTVβπ))βπ₯) β (π₯π»π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§))) |
18 | 17 | ralrimiva 3140 | . 2 β’ (π β βπ₯ β π (((Idβ(RngCatALTVβπ))βπ₯) β (π₯π»π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§))) |
19 | eqid 2733 | . . 3 β’ (Idβ(RngCatALTVβπ)) = (Idβ(RngCatALTVβπ)) | |
20 | eqid 2733 | . . 3 β’ (compβ(RngCatALTVβπ)) = (compβ(RngCatALTVβπ)) | |
21 | 7 | rngccatALTV 46378 | . . . 4 β’ (π β π β (RngCatALTVβπ) β Cat) |
22 | 1, 21 | syl 17 | . . 3 β’ (π β (RngCatALTVβπ) β Cat) |
23 | 1, 12, 2, 5 | rhmsubcALTVlem1 46492 | . . 3 β’ (π β π» Fn (π Γ π )) |
24 | 9, 19, 20, 22, 23 | issubc2 17730 | . 2 β’ (π β (π» β (Subcatβ(RngCatALTVβπ)) β (π» βcat (Homf β(RngCatALTVβπ)) β§ βπ₯ β π (((Idβ(RngCatALTVβπ))βπ₯) β (π₯π»π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β©(compβ(RngCatALTVβπ))π§)π) β (π₯π»π§))))) |
25 | 11, 18, 24 | mpbir2and 712 | 1 β’ (π β π» β (Subcatβ(RngCatALTVβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β© cin 3913 β¨cop 4596 class class class wbr 5109 Γ cxp 5635 βΎ cres 5639 βcfv 6500 (class class class)co 7361 compcco 17153 Catccat 17552 Idccid 17553 Homf chomf 17554 βcat cssc 17698 Subcatcsubc 17700 Ringcrg 19972 RingHom crh 20153 Rngcrng 46262 RngHomo crngh 46273 RngCatALTVcrngcALTV 46346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-hom 17165 df-cco 17166 df-0g 17331 df-cat 17556 df-cid 17557 df-homf 17558 df-ssc 17701 df-subc 17703 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-grp 18759 df-minusg 18760 df-ghm 19014 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-rnghom 20156 df-mgmhm 46163 df-rng 46263 df-rnghomo 46275 df-rngcALTV 46348 |
This theorem is referenced by: rhmsubcALTVcat 46497 |
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