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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringchom | Structured version Visualization version GIF version |
Description: Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) |
Ref | Expression |
---|---|
ringcbas.c | β’ πΆ = (RingCatβπ) |
ringcbas.b | β’ π΅ = (BaseβπΆ) |
ringcbas.u | β’ (π β π β π) |
ringchomfval.h | β’ π» = (Hom βπΆ) |
ringchom.x | β’ (π β π β π΅) |
ringchom.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
ringchom | β’ (π β (ππ»π) = (π RingHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcbas.c | . . . 4 β’ πΆ = (RingCatβπ) | |
2 | ringcbas.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
3 | ringcbas.u | . . . 4 β’ (π β π β π) | |
4 | ringchomfval.h | . . . 4 β’ π» = (Hom βπΆ) | |
5 | 1, 2, 3, 4 | ringchomfval 46989 | . . 3 β’ (π β π» = ( RingHom βΎ (π΅ Γ π΅))) |
6 | 5 | oveqd 7428 | . 2 β’ (π β (ππ»π) = (π( RingHom βΎ (π΅ Γ π΅))π)) |
7 | ringchom.x | . . 3 β’ (π β π β π΅) | |
8 | ringchom.y | . . 3 β’ (π β π β π΅) | |
9 | 7, 8 | ovresd 7576 | . 2 β’ (π β (π( RingHom βΎ (π΅ Γ π΅))π) = (π RingHom π)) |
10 | 6, 9 | eqtrd 2772 | 1 β’ (π β (ππ»π) = (π RingHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Γ cxp 5674 βΎ cres 5678 βcfv 6543 (class class class)co 7411 Basecbs 17146 Hom chom 17210 RingHom crh 20252 RingCatcringc 46980 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-hom 17223 df-cco 17224 df-0g 17389 df-resc 17760 df-estrc 18076 df-mhm 18673 df-ghm 19092 df-mgp 19990 df-ur 20007 df-ring 20060 df-rnghom 20255 df-ringc 46982 |
This theorem is referenced by: elringchom 46991 rhmsubcsetclem1 46998 rhmsubcrngclem1 47004 ringcsect 47008 funcringcsetc 47012 funcringcsetcALTV2lem8 47020 funcringcsetcALTV2lem9 47021 zrtermoringc 47047 zrninitoringc 47048 nzerooringczr 47049 srhmsubclem3 47052 srhmsubc 47053 |
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