![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ringcbasALTV | Structured version Visualization version GIF version |
Description: Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringcbasALTV.c | β’ πΆ = (RingCatALTVβπ) |
ringcbasALTV.b | β’ π΅ = (BaseβπΆ) |
ringcbasALTV.u | β’ (π β π β π) |
Ref | Expression |
---|---|
ringcbasALTV | β’ (π β π΅ = (π β© Ring)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcbasALTV.c | . . 3 β’ πΆ = (RingCatALTVβπ) | |
2 | ringcbasALTV.u | . . 3 β’ (π β π β π) | |
3 | eqidd 2738 | . . 3 β’ (π β (π β© Ring) = (π β© Ring)) | |
4 | eqidd 2738 | . . 3 β’ (π β (π₯ β (π β© Ring), π¦ β (π β© Ring) β¦ (π₯ RingHom π¦)) = (π₯ β (π β© Ring), π¦ β (π β© Ring) β¦ (π₯ RingHom π¦))) | |
5 | eqidd 2738 | . . 3 β’ (π β (π£ β ((π β© Ring) Γ (π β© Ring)), π§ β (π β© Ring) β¦ (π β ((2nd βπ£) RingHom π§), π β ((1st βπ£) RingHom (2nd βπ£)) β¦ (π β π))) = (π£ β ((π β© Ring) Γ (π β© Ring)), π§ β (π β© Ring) β¦ (π β ((2nd βπ£) RingHom π§), π β ((1st βπ£) RingHom (2nd βπ£)) β¦ (π β π)))) | |
6 | 1, 2, 3, 4, 5 | ringcvalALTV 46379 | . 2 β’ (π β πΆ = {β¨(Baseβndx), (π β© Ring)β©, β¨(Hom βndx), (π₯ β (π β© Ring), π¦ β (π β© Ring) β¦ (π₯ RingHom π¦))β©, β¨(compβndx), (π£ β ((π β© Ring) Γ (π β© Ring)), π§ β (π β© Ring) β¦ (π β ((2nd βπ£) RingHom π§), π β ((1st βπ£) RingHom (2nd βπ£)) β¦ (π β π)))β©}) |
7 | catstr 17852 | . 2 β’ {β¨(Baseβndx), (π β© Ring)β©, β¨(Hom βndx), (π₯ β (π β© Ring), π¦ β (π β© Ring) β¦ (π₯ RingHom π¦))β©, β¨(compβndx), (π£ β ((π β© Ring) Γ (π β© Ring)), π§ β (π β© Ring) β¦ (π β ((2nd βπ£) RingHom π§), π β ((1st βπ£) RingHom (2nd βπ£)) β¦ (π β π)))β©} Struct β¨1, ;15β© | |
8 | baseid 17093 | . 2 β’ Base = Slot (Baseβndx) | |
9 | snsstp1 4781 | . 2 β’ {β¨(Baseβndx), (π β© Ring)β©} β {β¨(Baseβndx), (π β© Ring)β©, β¨(Hom βndx), (π₯ β (π β© Ring), π¦ β (π β© Ring) β¦ (π₯ RingHom π¦))β©, β¨(compβndx), (π£ β ((π β© Ring) Γ (π β© Ring)), π§ β (π β© Ring) β¦ (π β ((2nd βπ£) RingHom π§), π β ((1st βπ£) RingHom (2nd βπ£)) β¦ (π β π)))β©} | |
10 | inex1g 5281 | . . 3 β’ (π β π β (π β© Ring) β V) | |
11 | 2, 10 | syl 17 | . 2 β’ (π β (π β© Ring) β V) |
12 | ringcbasALTV.b | . 2 β’ π΅ = (BaseβπΆ) | |
13 | 6, 7, 8, 9, 11, 12 | strfv3 17084 | 1 β’ (π β π΅ = (π β© Ring)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 β© cin 3914 {ctp 4595 β¨cop 4597 Γ cxp 5636 β ccom 5642 βcfv 6501 (class class class)co 7362 β cmpo 7364 1st c1st 7924 2nd c2nd 7925 1c1 11059 5c5 12218 ;cdc 12625 ndxcnx 17072 Basecbs 17090 Hom chom 17151 compcco 17152 Ringcrg 19971 RingHom crh 20152 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-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-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-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-slot 17061 df-ndx 17073 df-base 17091 df-hom 17164 df-cco 17165 df-ringcALTV 46378 |
This theorem is referenced by: ringchomfvalALTV 46419 ringccofvalALTV 46422 ringccatidALTV 46424 ringcbasbasALTV 46430 funcringcsetclem7ALTV 46437 srhmsubcALTVlem1 46464 srhmsubcALTV 46466 |
Copyright terms: Public domain | W3C validator |