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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 47191 | . . . . 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 17850 | . . . . 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 17300 | . . . . 5 β’ comp = Slot (compβndx) | |
7 | snsstp3 4779 | . . . . 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 5389 | . . . . . 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 2733 | . . . . 5 β’ (compβπΆ) = (compβπΆ) | |
11 | 4, 5, 6, 7, 9, 10 | strfv3 17082 | . . . 4 β’ (π β (compβπΆ) = {β¨β¨π, π, πβ©, (+gβπ)β©}) |
12 | 1, 11 | eqtrd 2773 | . . 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 47194 | . . . 4 β’ (π β π = π) |
16 | mndtchom.y | . . . . 5 β’ (π β π β π΅) | |
17 | 2, 3, 13, 16 | mndtcob 47194 | . . . 4 β’ (π β π = π) |
18 | 15, 17 | opeq12d 4839 | . . 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 47194 | . . 3 β’ (π β π = π) |
21 | 12, 18, 20 | oveq123d 7379 | . 2 β’ (π β (β¨π, πβ© Β· π) = (β¨π, πβ©{β¨β¨π, π, πβ©, (+gβπ)β©}π)) |
22 | df-ov 7361 | . . 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 4596 | . . . 4 β’ β¨π, π, πβ© = β¨β¨π, πβ©, πβ© | |
24 | 23 | fveq2i 6846 | . . 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 5423 | . . . 4 β’ β¨π, π, πβ© β V | |
26 | fvex 6856 | . . . 4 β’ (+gβπ) β V | |
27 | 25, 26 | fvsn 7128 | . . 3 β’ ({β¨β¨π, π, πβ©, (+gβπ)β©}ββ¨π, π, πβ©) = (+gβπ) |
28 | 22, 24, 27 | 3eqtr2i 2767 | . 2 β’ (β¨π, πβ©{β¨β¨π, π, πβ©, (+gβπ)β©}π) = (+gβπ) |
29 | 21, 28 | eqtrdi 2789 | 1 β’ (π β (β¨π, πβ© Β· π) = (+gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 {csn 4587 {ctp 4591 β¨cop 4593 β¨cotp 4595 βcfv 6497 (class class class)co 7358 1c1 11057 5c5 12216 ;cdc 12623 ndxcnx 17070 Basecbs 17088 +gcplusg 17138 Hom chom 17149 compcco 17150 Mndcmnd 18561 MndToCatcmndtc 47189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-mndtc 47190 |
This theorem is referenced by: mndtcco2 47198 mndtccatid 47199 |
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