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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldcALTV | Structured version Visualization version GIF version |
Description: The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
drhmsubcALTV.c | β’ πΆ = (π β© DivRing) |
drhmsubcALTV.j | β’ π½ = (π β πΆ, π β πΆ β¦ (π RingHom π )) |
fldhmsubcALTV.d | β’ π· = (π β© Field) |
fldhmsubcALTV.f | β’ πΉ = (π β π·, π β π· β¦ (π RingHom π )) |
Ref | Expression |
---|---|
fldcALTV | β’ (π β π β (((RingCatALTVβπ) βΎcat π½) βΎcat πΉ) β Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6862 | . . 3 β’ (π β π β (RingCatALTVβπ) β V) | |
2 | drhmsubcALTV.j | . . . . 5 β’ π½ = (π β πΆ, π β πΆ β¦ (π RingHom π )) | |
3 | ovex 7395 | . . . . 5 β’ (π RingHom π ) β V | |
4 | 2, 3 | fnmpoi 8007 | . . . 4 β’ π½ Fn (πΆ Γ πΆ) |
5 | 4 | a1i 11 | . . 3 β’ (π β π β π½ Fn (πΆ Γ πΆ)) |
6 | fldhmsubcALTV.f | . . . . 5 β’ πΉ = (π β π·, π β π· β¦ (π RingHom π )) | |
7 | 6, 3 | fnmpoi 8007 | . . . 4 β’ πΉ Fn (π· Γ π·) |
8 | 7 | a1i 11 | . . 3 β’ (π β π β πΉ Fn (π· Γ π·)) |
9 | drhmsubcALTV.c | . . . 4 β’ πΆ = (π β© DivRing) | |
10 | inex1g 5281 | . . . 4 β’ (π β π β (π β© DivRing) β V) | |
11 | 9, 10 | eqeltrid 2842 | . . 3 β’ (π β π β πΆ β V) |
12 | df-field 20202 | . . . . . 6 β’ Field = (DivRing β© CRing) | |
13 | inss1 4193 | . . . . . 6 β’ (DivRing β© CRing) β DivRing | |
14 | 12, 13 | eqsstri 3983 | . . . . 5 β’ Field β DivRing |
15 | sslin 4199 | . . . . 5 β’ (Field β DivRing β (π β© Field) β (π β© DivRing)) | |
16 | 14, 15 | mp1i 13 | . . . 4 β’ (π β π β (π β© Field) β (π β© DivRing)) |
17 | fldhmsubcALTV.d | . . . 4 β’ π· = (π β© Field) | |
18 | 16, 17, 9 | 3sstr4g 3994 | . . 3 β’ (π β π β π· β πΆ) |
19 | 1, 5, 8, 11, 18 | rescabs 17725 | . 2 β’ (π β π β (((RingCatALTVβπ) βΎcat π½) βΎcat πΉ) = ((RingCatALTVβπ) βΎcat πΉ)) |
20 | 9, 2, 17, 6 | fldcatALTV 46472 | . 2 β’ (π β π β ((RingCatALTVβπ) βΎcat πΉ) β Cat) |
21 | 19, 20 | eqeltrd 2838 | 1 β’ (π β π β (((RingCatALTVβπ) βΎcat π½) βΎcat πΉ) β Cat) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 β© cin 3914 β wss 3915 Γ cxp 5636 Fn wfn 6496 βcfv 6501 (class class class)co 7362 β cmpo 7364 Catccat 17551 βΎcat cresc 17698 CRingccrg 19972 RingHom crh 20152 DivRingcdr 20199 Fieldcfield 20200 RingCatALTVcringcALTV 46376 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-hom 17164 df-cco 17165 df-0g 17330 df-cat 17555 df-cid 17556 df-homf 17557 df-ssc 17700 df-resc 17701 df-subc 17702 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-ghm 19013 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-rnghom 20155 df-field 20202 df-ringcALTV 46378 |
This theorem is referenced by: (None) |
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