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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 46910 | . . 3 β’ (π β π» = ( RingHom βΎ (π΅ Γ π΅))) |
6 | 5 | oveqd 7426 | . 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 7574 | . 2 β’ (π β (π( RingHom βΎ (π΅ Γ π΅))π) = (π RingHom π)) |
10 | 6, 9 | eqtrd 2773 | 1 β’ (π β (ππ»π) = (π RingHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Γ cxp 5675 βΎ cres 5679 βcfv 6544 (class class class)co 7409 Basecbs 17144 Hom chom 17208 RingHom crh 20248 RingCatcringc 46901 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-hom 17221 df-cco 17222 df-0g 17387 df-resc 17758 df-estrc 18074 df-mhm 18671 df-ghm 19090 df-mgp 19988 df-ur 20005 df-ring 20058 df-rnghom 20251 df-ringc 46903 |
This theorem is referenced by: elringchom 46912 rhmsubcsetclem1 46919 rhmsubcrngclem1 46925 ringcsect 46929 funcringcsetc 46933 funcringcsetcALTV2lem8 46941 funcringcsetcALTV2lem9 46942 zrtermoringc 46968 zrninitoringc 46969 nzerooringczr 46970 srhmsubclem3 46973 srhmsubc 46974 |
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