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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcid | Structured version Visualization version GIF version |
Description: The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
Ref | Expression |
---|---|
rngccat.c | β’ πΆ = (RngCatβπ) |
rngcid.b | β’ π΅ = (BaseβπΆ) |
rngcid.o | β’ 1 = (IdβπΆ) |
rngcid.u | β’ (π β π β π) |
rngcid.x | β’ (π β π β π΅) |
rngcid.s | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
rngcid | β’ (π β ( 1 βπ) = ( I βΎ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcid.o | . . . 4 β’ 1 = (IdβπΆ) | |
2 | rngccat.c | . . . . . 6 β’ πΆ = (RngCatβπ) | |
3 | rngcid.u | . . . . . 6 β’ (π β π β π) | |
4 | eqidd 2732 | . . . . . 6 β’ (π β (π β© Rng) = (π β© Rng)) | |
5 | eqidd 2732 | . . . . . 6 β’ (π β ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng))) = ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))) | |
6 | 2, 3, 4, 5 | rngcval 46413 | . . . . 5 β’ (π β πΆ = ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng))))) |
7 | 6 | fveq2d 6866 | . . . 4 β’ (π β (IdβπΆ) = (Idβ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))))) |
8 | 1, 7 | eqtrid 2783 | . . 3 β’ (π β 1 = (Idβ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))))) |
9 | 8 | fveq1d 6864 | . 2 β’ (π β ( 1 βπ) = ((Idβ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))))βπ)) |
10 | eqid 2731 | . . 3 β’ ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))) = ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))) | |
11 | eqid 2731 | . . . 4 β’ (ExtStrCatβπ) = (ExtStrCatβπ) | |
12 | incom 4181 | . . . . 5 β’ (π β© Rng) = (Rng β© π) | |
13 | 12 | a1i 11 | . . . 4 β’ (π β (π β© Rng) = (Rng β© π)) |
14 | 11, 3, 13, 5 | rnghmsubcsetc 46428 | . . 3 β’ (π β ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng))) β (Subcatβ(ExtStrCatβπ))) |
15 | 4, 5 | rnghmresfn 46414 | . . 3 β’ (π β ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng))) Fn ((π β© Rng) Γ (π β© Rng))) |
16 | eqid 2731 | . . 3 β’ (Idβ(ExtStrCatβπ)) = (Idβ(ExtStrCatβπ)) | |
17 | rngcid.x | . . . 4 β’ (π β π β π΅) | |
18 | rngcid.b | . . . . . 6 β’ π΅ = (BaseβπΆ) | |
19 | 2, 18, 3 | rngcbas 46416 | . . . . 5 β’ (π β π΅ = (π β© Rng)) |
20 | 19 | eleq2d 2818 | . . . 4 β’ (π β (π β π΅ β π β (π β© Rng))) |
21 | 17, 20 | mpbid 231 | . . 3 β’ (π β π β (π β© Rng)) |
22 | 10, 14, 15, 16, 21 | subcid 17762 | . 2 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ((Idβ((ExtStrCatβπ) βΎcat ( RngHomo βΎ ((π β© Rng) Γ (π β© Rng)))))βπ)) |
23 | elinel1 4175 | . . . . . 6 β’ (π β (π β© Rng) β π β π) | |
24 | 20, 23 | syl6bi 252 | . . . . 5 β’ (π β (π β π΅ β π β π)) |
25 | 17, 24 | mpd 15 | . . . 4 β’ (π β π β π) |
26 | 11, 16, 3, 25 | estrcid 18050 | . . 3 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ( I βΎ (Baseβπ))) |
27 | rngcid.s | . . . . . 6 β’ π = (Baseβπ) | |
28 | 27 | eqcomi 2740 | . . . . 5 β’ (Baseβπ) = π |
29 | 28 | a1i 11 | . . . 4 β’ (π β (Baseβπ) = π) |
30 | 29 | reseq2d 5957 | . . 3 β’ (π β ( I βΎ (Baseβπ)) = ( I βΎ π)) |
31 | 26, 30 | eqtrd 2771 | . 2 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ( I βΎ π)) |
32 | 9, 22, 31 | 3eqtr2d 2777 | 1 β’ (π β ( 1 βπ) = ( I βΎ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β© cin 3927 I cid 5550 Γ cxp 5651 βΎ cres 5655 βcfv 6516 (class class class)co 7377 Basecbs 17109 Idccid 17574 βΎcat cresc 17720 ExtStrCatcestrc 18038 Rngcrng 46325 RngHomo crngh 46336 RngCatcrngc 46408 |
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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-map 8789 df-pm 8790 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-fz 13450 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-hom 17186 df-cco 17187 df-0g 17352 df-cat 17577 df-cid 17578 df-homf 17579 df-ssc 17722 df-resc 17723 df-subc 17724 df-estrc 18039 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-mhm 18630 df-grp 18780 df-ghm 19035 df-abl 19594 df-mgp 19926 df-mgmhm 46226 df-rng 46326 df-rnghomo 46338 df-rngc 46410 |
This theorem is referenced by: rngcsect 46431 rhmsubcrngclem1 46478 rhmsubclem3 46539 |
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