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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcifuestrc | Structured version Visualization version GIF version |
Description: The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.) |
Ref | Expression |
---|---|
rngcifuestrc.r | β’ π = (RngCatβπ) |
rngcifuestrc.e | β’ πΈ = (ExtStrCatβπ) |
rngcifuestrc.b | β’ π΅ = (Baseβπ ) |
rngcifuestrc.u | β’ (π β π β π) |
rngcifuestrc.f | β’ (π β πΉ = ( I βΎ π΅)) |
rngcifuestrc.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RngHomo π¦)))) |
Ref | Expression |
---|---|
rngcifuestrc | β’ (π β πΉ(π Func πΈ)πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . . 5 β’ (ExtStrCatβπ) = (ExtStrCatβπ) | |
2 | rngcifuestrc.u | . . . . 5 β’ (π β π β π) | |
3 | rngcifuestrc.r | . . . . . . 7 β’ π = (RngCatβπ) | |
4 | rngcifuestrc.b | . . . . . . 7 β’ π΅ = (Baseβπ ) | |
5 | 3, 4, 2 | rngcbas 46853 | . . . . . 6 β’ (π β π΅ = (π β© Rng)) |
6 | incom 4201 | . . . . . 6 β’ (π β© Rng) = (Rng β© π) | |
7 | 5, 6 | eqtrdi 2788 | . . . . 5 β’ (π β π΅ = (Rng β© π)) |
8 | eqid 2732 | . . . . . 6 β’ (Hom βπ ) = (Hom βπ ) | |
9 | 3, 4, 2, 8 | rngchomfval 46854 | . . . . 5 β’ (π β (Hom βπ ) = ( RngHomo βΎ (π΅ Γ π΅))) |
10 | 1, 2, 7, 9 | rnghmsubcsetc 46865 | . . . 4 β’ (π β (Hom βπ ) β (Subcatβ(ExtStrCatβπ))) |
11 | 10 | idi 1 | . . 3 β’ (π β (Hom βπ ) β (Subcatβ(ExtStrCatβπ))) |
12 | eqid 2732 | . . 3 β’ ((ExtStrCatβπ) βΎcat (Hom βπ )) = ((ExtStrCatβπ) βΎcat (Hom βπ )) | |
13 | eqid 2732 | . . 3 β’ (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))) = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))) | |
14 | rngcifuestrc.f | . . . 4 β’ (π β πΉ = ( I βΎ π΅)) | |
15 | 3, 2, 5, 9 | rngcval 46850 | . . . . . . 7 β’ (π β π = ((ExtStrCatβπ) βΎcat (Hom βπ ))) |
16 | 15 | fveq2d 6895 | . . . . . 6 β’ (π β (Baseβπ ) = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
17 | 4, 16 | eqtrid 2784 | . . . . 5 β’ (π β π΅ = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
18 | 17 | reseq2d 5981 | . . . 4 β’ (π β ( I βΎ π΅) = ( I βΎ (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))))) |
19 | 14, 18 | eqtrd 2772 | . . 3 β’ (π β πΉ = ( I βΎ (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))))) |
20 | rngcifuestrc.g | . . . 4 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RngHomo π¦)))) | |
21 | 17 | adantr 481 | . . . . 5 β’ ((π β§ π₯ β π΅) β π΅ = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
22 | 9 | oveqdr 7436 | . . . . . . 7 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(Hom βπ )π¦) = (π₯( RngHomo βΎ (π΅ Γ π΅))π¦)) |
23 | ovres 7572 | . . . . . . . 8 β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯( RngHomo βΎ (π΅ Γ π΅))π¦) = (π₯ RngHomo π¦)) | |
24 | 23 | adantl 482 | . . . . . . 7 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯( RngHomo βΎ (π΅ Γ π΅))π¦) = (π₯ RngHomo π¦)) |
25 | 22, 24 | eqtr2d 2773 | . . . . . 6 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯ RngHomo π¦) = (π₯(Hom βπ )π¦)) |
26 | 25 | reseq2d 5981 | . . . . 5 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β ( I βΎ (π₯ RngHomo π¦)) = ( I βΎ (π₯(Hom βπ )π¦))) |
27 | 17, 21, 26 | mpoeq123dva 7482 | . . . 4 β’ (π β (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RngHomo π¦))) = (π₯ β (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))), π¦ β (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))) β¦ ( I βΎ (π₯(Hom βπ )π¦)))) |
28 | 20, 27 | eqtrd 2772 | . . 3 β’ (π β πΊ = (π₯ β (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))), π¦ β (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))) β¦ ( I βΎ (π₯(Hom βπ )π¦)))) |
29 | 11, 12, 13, 19, 28 | inclfusubc 46631 | . 2 β’ (π β πΉ(((ExtStrCatβπ) βΎcat (Hom βπ )) Func (ExtStrCatβπ))πΊ) |
30 | rngcifuestrc.e | . . . . 5 β’ πΈ = (ExtStrCatβπ) | |
31 | 30 | a1i 11 | . . . 4 β’ (π β πΈ = (ExtStrCatβπ)) |
32 | 15, 31 | oveq12d 7426 | . . 3 β’ (π β (π Func πΈ) = (((ExtStrCatβπ) βΎcat (Hom βπ )) Func (ExtStrCatβπ))) |
33 | 32 | breqd 5159 | . 2 β’ (π β (πΉ(π Func πΈ)πΊ β πΉ(((ExtStrCatβπ) βΎcat (Hom βπ )) Func (ExtStrCatβπ))πΊ)) |
34 | 29, 33 | mpbird 256 | 1 β’ (π β πΉ(π Func πΈ)πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β© cin 3947 class class class wbr 5148 I cid 5573 Γ cxp 5674 βΎ cres 5678 βcfv 6543 (class class class)co 7408 β cmpo 7410 Basecbs 17143 Hom chom 17207 βΎcat cresc 17754 Subcatcsubc 17755 Func cfunc 17803 ExtStrCatcestrc 18072 Rngcrng 46638 RngHomo crngh 46673 RngCatcrngc 46845 |
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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-rmo 3376 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-hom 17220 df-cco 17221 df-0g 17386 df-cat 17611 df-cid 17612 df-homf 17613 df-ssc 17756 df-resc 17757 df-subc 17758 df-func 17807 df-idfu 17808 df-full 17854 df-fth 17855 df-estrc 18073 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-grp 18821 df-ghm 19089 df-abl 19650 df-mgp 19987 df-mgmhm 46539 df-rng 46639 df-rnghomo 46675 df-rngc 46847 |
This theorem is referenced by: funcrngcsetcALT 46887 |
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