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Mirrors > Home > MPE Home > Th. List > ringcid | Structured version Visualization version GIF version |
Description: The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
Ref | Expression |
---|---|
ringccat.c | β’ πΆ = (RingCatβπ) |
ringcid.b | β’ π΅ = (BaseβπΆ) |
ringcid.o | β’ 1 = (IdβπΆ) |
ringcid.u | β’ (π β π β π) |
ringcid.x | β’ (π β π β π΅) |
ringcid.s | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
ringcid | β’ (π β ( 1 βπ) = ( I βΎ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcid.o | . . . 4 β’ 1 = (IdβπΆ) | |
2 | ringccat.c | . . . . . 6 β’ πΆ = (RingCatβπ) | |
3 | ringcid.u | . . . . . 6 β’ (π β π β π) | |
4 | eqidd 2729 | . . . . . 6 β’ (π β (π β© Ring) = (π β© Ring)) | |
5 | eqidd 2729 | . . . . . 6 β’ (π β ( RingHom βΎ ((π β© Ring) Γ (π β© Ring))) = ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))) | |
6 | 2, 3, 4, 5 | ringcval 20587 | . . . . 5 β’ (π β πΆ = ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring))))) |
7 | 6 | fveq2d 6906 | . . . 4 β’ (π β (IdβπΆ) = (Idβ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))))) |
8 | 1, 7 | eqtrid 2780 | . . 3 β’ (π β 1 = (Idβ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))))) |
9 | 8 | fveq1d 6904 | . 2 β’ (π β ( 1 βπ) = ((Idβ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))))βπ)) |
10 | eqid 2728 | . . 3 β’ ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))) = ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))) | |
11 | eqid 2728 | . . . 4 β’ (ExtStrCatβπ) = (ExtStrCatβπ) | |
12 | incom 4203 | . . . . 5 β’ (π β© Ring) = (Ring β© π) | |
13 | 12 | a1i 11 | . . . 4 β’ (π β (π β© Ring) = (Ring β© π)) |
14 | 11, 3, 13, 5 | rhmsubcsetc 20602 | . . 3 β’ (π β ( RingHom βΎ ((π β© Ring) Γ (π β© Ring))) β (Subcatβ(ExtStrCatβπ))) |
15 | 4, 5 | rhmresfn 20588 | . . 3 β’ (π β ( RingHom βΎ ((π β© Ring) Γ (π β© Ring))) Fn ((π β© Ring) Γ (π β© Ring))) |
16 | eqid 2728 | . . 3 β’ (Idβ(ExtStrCatβπ)) = (Idβ(ExtStrCatβπ)) | |
17 | ringcid.x | . . . 4 β’ (π β π β π΅) | |
18 | ringcid.b | . . . . . 6 β’ π΅ = (BaseβπΆ) | |
19 | 2, 18, 3 | ringcbas 20590 | . . . . 5 β’ (π β π΅ = (π β© Ring)) |
20 | 19 | eleq2d 2815 | . . . 4 β’ (π β (π β π΅ β π β (π β© Ring))) |
21 | 17, 20 | mpbid 231 | . . 3 β’ (π β π β (π β© Ring)) |
22 | 10, 14, 15, 16, 21 | subcid 17840 | . 2 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ((Idβ((ExtStrCatβπ) βΎcat ( RingHom βΎ ((π β© Ring) Γ (π β© Ring)))))βπ)) |
23 | elinel1 4197 | . . . . . 6 β’ (π β (π β© Ring) β π β π) | |
24 | 20, 23 | biimtrdi 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 18131 | . . 3 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ( I βΎ (Baseβπ))) |
27 | ringcid.s | . . . . . 6 β’ π = (Baseβπ) | |
28 | 27 | eqcomi 2737 | . . . . 5 β’ (Baseβπ) = π |
29 | 28 | a1i 11 | . . . 4 β’ (π β (Baseβπ) = π) |
30 | 29 | reseq2d 5989 | . . 3 β’ (π β ( I βΎ (Baseβπ)) = ( I βΎ π)) |
31 | 26, 30 | eqtrd 2768 | . 2 β’ (π β ((Idβ(ExtStrCatβπ))βπ) = ( I βΎ π)) |
32 | 9, 22, 31 | 3eqtr2d 2774 | 1 β’ (π β ( 1 βπ) = ( I βΎ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β© cin 3948 I cid 5579 Γ cxp 5680 βΎ cres 5684 βcfv 6553 (class class class)co 7426 Basecbs 17187 Idccid 17652 βΎcat cresc 17798 ExtStrCatcestrc 18119 Ringcrg 20180 RingHom crh 20415 RingCatcringc 20585 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-hom 17264 df-cco 17265 df-0g 17430 df-cat 17655 df-cid 17656 df-homf 17657 df-ssc 17800 df-resc 17801 df-subc 17802 df-estrc 18120 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-ghm 19175 df-mgp 20082 df-ur 20129 df-ring 20182 df-rhm 20418 df-ringc 20586 |
This theorem is referenced by: ringcsect 20610 srhmsubc 20620 funcringcsetcALTV2lem7 47436 |
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