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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndtcco | Structured version Visualization version GIF version |
Description: The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
Ref | Expression |
---|---|
mndtcbas.c | β’ (π β πΆ = (MndToCatβπ)) |
mndtcbas.m | β’ (π β π β Mnd) |
mndtcbas.b | β’ (π β π΅ = (BaseβπΆ)) |
mndtchom.x | β’ (π β π β π΅) |
mndtchom.y | β’ (π β π β π΅) |
mndtcco.z | β’ (π β π β π΅) |
mndtcco.o | β’ (π β Β· = (compβπΆ)) |
Ref | Expression |
---|---|
mndtcco | β’ (π β (β¨π, πβ© Β· π) = (+gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndtcco.o | . . . 4 β’ (π β Β· = (compβπΆ)) | |
2 | mndtcbas.c | . . . . . 6 β’ (π β πΆ = (MndToCatβπ)) | |
3 | mndtcbas.m | . . . . . 6 β’ (π β π β Mnd) | |
4 | 2, 3 | mndtcval 48169 | . . . . 5 β’ (π β πΆ = {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©}) |
5 | catstr 17955 | . . . . 5 β’ {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©} Struct β¨1, ;15β© | |
6 | ccoid 17402 | . . . . 5 β’ comp = Slot (compβndx) | |
7 | snsstp3 4826 | . . . . 5 β’ {β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©} β {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©} | |
8 | snex 5437 | . . . . . 6 β’ {β¨β¨π, π, πβ©, (+gβπ)β©} β V | |
9 | 8 | a1i 11 | . . . . 5 β’ (π β {β¨β¨π, π, πβ©, (+gβπ)β©} β V) |
10 | eqid 2728 | . . . . 5 β’ (compβπΆ) = (compβπΆ) | |
11 | 4, 5, 6, 7, 9, 10 | strfv3 17181 | . . . 4 β’ (π β (compβπΆ) = {β¨β¨π, π, πβ©, (+gβπ)β©}) |
12 | 1, 11 | eqtrd 2768 | . . 3 β’ (π β Β· = {β¨β¨π, π, πβ©, (+gβπ)β©}) |
13 | mndtcbas.b | . . . . 5 β’ (π β π΅ = (BaseβπΆ)) | |
14 | mndtchom.x | . . . . 5 β’ (π β π β π΅) | |
15 | 2, 3, 13, 14 | mndtcob 48172 | . . . 4 β’ (π β π = π) |
16 | mndtchom.y | . . . . 5 β’ (π β π β π΅) | |
17 | 2, 3, 13, 16 | mndtcob 48172 | . . . 4 β’ (π β π = π) |
18 | 15, 17 | opeq12d 4886 | . . 3 β’ (π β β¨π, πβ© = β¨π, πβ©) |
19 | mndtcco.z | . . . 4 β’ (π β π β π΅) | |
20 | 2, 3, 13, 19 | mndtcob 48172 | . . 3 β’ (π β π = π) |
21 | 12, 18, 20 | oveq123d 7447 | . 2 β’ (π β (β¨π, πβ© Β· π) = (β¨π, πβ©{β¨β¨π, π, πβ©, (+gβπ)β©}π)) |
22 | df-ov 7429 | . . 3 β’ (β¨π, πβ©{β¨β¨π, π, πβ©, (+gβπ)β©}π) = ({β¨β¨π, π, πβ©, (+gβπ)β©}ββ¨β¨π, πβ©, πβ©) | |
23 | df-ot 4641 | . . . 4 β’ β¨π, π, πβ© = β¨β¨π, πβ©, πβ© | |
24 | 23 | fveq2i 6905 | . . 3 β’ ({β¨β¨π, π, πβ©, (+gβπ)β©}ββ¨π, π, πβ©) = ({β¨β¨π, π, πβ©, (+gβπ)β©}ββ¨β¨π, πβ©, πβ©) |
25 | otex 5471 | . . . 4 β’ β¨π, π, πβ© β V | |
26 | fvex 6915 | . . . 4 β’ (+gβπ) β V | |
27 | 25, 26 | fvsn 7196 | . . 3 β’ ({β¨β¨π, π, πβ©, (+gβπ)β©}ββ¨π, π, πβ©) = (+gβπ) |
28 | 22, 24, 27 | 3eqtr2i 2762 | . 2 β’ (β¨π, πβ©{β¨β¨π, π, πβ©, (+gβπ)β©}π) = (+gβπ) |
29 | 21, 28 | eqtrdi 2784 | 1 β’ (π β (β¨π, πβ© Β· π) = (+gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3473 {csn 4632 {ctp 4636 β¨cop 4638 β¨cotp 4640 βcfv 6553 (class class class)co 7426 1c1 11147 5c5 12308 ;cdc 12715 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 Hom chom 17251 compcco 17252 Mndcmnd 18701 MndToCatcmndtc 48167 |
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-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-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-ot 4641 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-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-slot 17158 df-ndx 17170 df-base 17188 df-hom 17264 df-cco 17265 df-mndtc 48168 |
This theorem is referenced by: mndtcco2 48176 mndtccatid 48177 |
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