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Mathbox for Alexander van der Vekens |
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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 2728 | . . 3 β’ (π β (π β© Ring) = (π β© Ring)) | |
4 | eqidd 2728 | . . 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 2728 | . . 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 47274 | . 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 17939 | . 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 17174 | . 2 β’ Base = Slot (Baseβndx) | |
9 | snsstp1 4815 | . 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 5313 | . . 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 17165 | 1 β’ (π β π΅ = (π β© Ring)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3469 β© cin 3943 {ctp 4628 β¨cop 4630 Γ cxp 5670 β ccom 5676 βcfv 6542 (class class class)co 7414 β cmpo 7416 1st c1st 7985 2nd c2nd 7986 1c1 11131 5c5 12292 ;cdc 12699 ndxcnx 17153 Basecbs 17171 Hom chom 17235 compcco 17236 Ringcrg 20164 RingHom crh 20397 RingCatALTVcringcALTV 47272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-hom 17248 df-cco 17249 df-ringcALTV 47273 |
This theorem is referenced by: ringchomfvalALTV 47286 ringccofvalALTV 47289 ringccatidALTV 47291 ringcbasbasALTV 47297 funcringcsetclem7ALTV 47304 srhmsubcALTVlem1 47308 srhmsubcALTV 47310 |
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