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Mirrors > Home > MPE Home > Th. List > 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₯ RngHom π¦)))) |
Ref | Expression |
---|---|
rngcifuestrc | β’ (π β πΉ(π Func πΈ)πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 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 20517 | . . . . . 6 β’ (π β π΅ = (π β© Rng)) |
6 | incom 4196 | . . . . . 6 β’ (π β© Rng) = (Rng β© π) | |
7 | 5, 6 | eqtrdi 2782 | . . . . 5 β’ (π β π΅ = (Rng β© π)) |
8 | eqid 2726 | . . . . . 6 β’ (Hom βπ ) = (Hom βπ ) | |
9 | 3, 4, 2, 8 | rngchomfval 20518 | . . . . 5 β’ (π β (Hom βπ ) = ( RngHom βΎ (π΅ Γ π΅))) |
10 | 1, 2, 7, 9 | rnghmsubcsetc 20529 | . . . 4 β’ (π β (Hom βπ ) β (Subcatβ(ExtStrCatβπ))) |
11 | 10 | idi 1 | . . 3 β’ (π β (Hom βπ ) β (Subcatβ(ExtStrCatβπ))) |
12 | eqid 2726 | . . 3 β’ ((ExtStrCatβπ) βΎcat (Hom βπ )) = ((ExtStrCatβπ) βΎcat (Hom βπ )) | |
13 | eqid 2726 | . . 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 20514 | . . . . . . 7 β’ (π β π = ((ExtStrCatβπ) βΎcat (Hom βπ ))) |
16 | 15 | fveq2d 6889 | . . . . . 6 β’ (π β (Baseβπ ) = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
17 | 4, 16 | eqtrid 2778 | . . . . 5 β’ (π β π΅ = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
18 | 17 | reseq2d 5975 | . . . 4 β’ (π β ( I βΎ π΅) = ( I βΎ (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ ))))) |
19 | 14, 18 | eqtrd 2766 | . . 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₯ RngHom π¦)))) | |
21 | 17 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π΅) β π΅ = (Baseβ((ExtStrCatβπ) βΎcat (Hom βπ )))) |
22 | 9 | oveqdr 7433 | . . . . . . 7 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(Hom βπ )π¦) = (π₯( RngHom βΎ (π΅ Γ π΅))π¦)) |
23 | ovres 7570 | . . . . . . . 8 β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯( RngHom βΎ (π΅ Γ π΅))π¦) = (π₯ RngHom π¦)) | |
24 | 23 | adantl 481 | . . . . . . 7 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯( RngHom βΎ (π΅ Γ π΅))π¦) = (π₯ RngHom π¦)) |
25 | 22, 24 | eqtr2d 2767 | . . . . . 6 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯ RngHom π¦) = (π₯(Hom βπ )π¦)) |
26 | 25 | reseq2d 5975 | . . . . 5 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β ( I βΎ (π₯ RngHom π¦)) = ( I βΎ (π₯(Hom βπ )π¦))) |
27 | 17, 21, 26 | mpoeq123dva 7479 | . . . 4 β’ (π β (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RngHom 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 2766 | . . 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 17903 | . 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 7423 | . . 3 β’ (π β (π Func πΈ) = (((ExtStrCatβπ) βΎcat (Hom βπ )) Func (ExtStrCatβπ))) |
33 | 32 | breqd 5152 | . 2 β’ (π β (πΉ(π Func πΈ)πΊ β πΉ(((ExtStrCatβπ) βΎcat (Hom βπ )) Func (ExtStrCatβπ))πΊ)) |
34 | 29, 33 | mpbird 257 | 1 β’ (π β πΉ(π Func πΈ)πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β© cin 3942 class class class wbr 5141 I cid 5566 Γ cxp 5667 βΎ cres 5671 βcfv 6537 (class class class)co 7405 β cmpo 7407 Basecbs 17153 Hom chom 17217 βΎcat cresc 17764 Subcatcsubc 17765 Func cfunc 17813 ExtStrCatcestrc 18085 Rngcrng 20057 RngHom crnghm 20336 RngCatcrngc 20512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-hom 17230 df-cco 17231 df-0g 17396 df-cat 17621 df-cid 17622 df-homf 17623 df-ssc 17766 df-resc 17767 df-subc 17768 df-func 17817 df-idfu 17818 df-full 17866 df-fth 17867 df-estrc 18086 df-mgm 18573 df-mgmhm 18625 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-ghm 19139 df-abl 19703 df-mgp 20040 df-rng 20058 df-rnghm 20338 df-rngc 20513 |
This theorem is referenced by: funcrngcsetcALT 20537 |
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