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Mirrors > Home > MPE Home > Th. List > smndex1sgrp | Structured version Visualization version GIF version |
Description: The monoid of endofunctions on β0 restricted to the modulo function πΌ and the constant functions (πΊβπΎ) is a semigroup. (Contributed by AV, 14-Feb-2024.) |
Ref | Expression |
---|---|
smndex1ibas.m | β’ π = (EndoFMndββ0) |
smndex1ibas.n | β’ π β β |
smndex1ibas.i | β’ πΌ = (π₯ β β0 β¦ (π₯ mod π)) |
smndex1ibas.g | β’ πΊ = (π β (0..^π) β¦ (π₯ β β0 β¦ π)) |
smndex1mgm.b | β’ π΅ = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) |
smndex1mgm.s | β’ π = (π βΎs π΅) |
Ref | Expression |
---|---|
smndex1sgrp | β’ π β Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.m | . . 3 β’ π = (EndoFMndββ0) | |
2 | smndex1ibas.n | . . 3 β’ π β β | |
3 | smndex1ibas.i | . . 3 β’ πΌ = (π₯ β β0 β¦ (π₯ mod π)) | |
4 | smndex1ibas.g | . . 3 β’ πΊ = (π β (0..^π) β¦ (π₯ β β0 β¦ π)) | |
5 | smndex1mgm.b | . . 3 β’ π΅ = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) | |
6 | smndex1mgm.s | . . 3 β’ π = (π βΎs π΅) | |
7 | 1, 2, 3, 4, 5, 6 | smndex1mgm 18718 | . 2 β’ π β Mgm |
8 | eqid 2737 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
9 | eqid 2737 | . . . . . 6 β’ (+gβπ) = (+gβπ) | |
10 | 8, 9 | mgmcl 18501 | . . . . 5 β’ ((π β Mgm β§ π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) β (Baseβπ)) |
11 | 7, 10 | mp3an1 1449 | . . . 4 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) β (Baseβπ)) |
12 | snex 5389 | . . . . . . . . . 10 β’ {πΌ} β V | |
13 | ovex 7391 | . . . . . . . . . . 11 β’ (0..^π) β V | |
14 | snex 5389 | . . . . . . . . . . 11 β’ {(πΊβπ)} β V | |
15 | 13, 14 | iunex 7902 | . . . . . . . . . 10 β’ βͺ π β (0..^π){(πΊβπ)} β V |
16 | 12, 15 | unex 7681 | . . . . . . . . 9 β’ ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) β V |
17 | 5, 16 | eqeltri 2834 | . . . . . . . 8 β’ π΅ β V |
18 | eqid 2737 | . . . . . . . . 9 β’ (+gβπ) = (+gβπ) | |
19 | 6, 18 | ressplusg 17172 | . . . . . . . 8 β’ (π΅ β V β (+gβπ) = (+gβπ)) |
20 | 17, 19 | ax-mp 5 | . . . . . . 7 β’ (+gβπ) = (+gβπ) |
21 | 20 | eqcomi 2746 | . . . . . 6 β’ (+gβπ) = (+gβπ) |
22 | 21 | oveqi 7371 | . . . . 5 β’ (π₯(+gβπ)π¦) = (π₯(+gβπ)π¦) |
23 | 1, 2, 3, 4, 5, 6 | smndex1bas 18717 | . . . . . . . 8 β’ (Baseβπ) = π΅ |
24 | 1, 2, 3, 4, 5 | smndex1basss 18716 | . . . . . . . 8 β’ π΅ β (Baseβπ) |
25 | 23, 24 | eqsstri 3979 | . . . . . . 7 β’ (Baseβπ) β (Baseβπ) |
26 | ssel 3938 | . . . . . . . 8 β’ ((Baseβπ) β (Baseβπ) β (π₯ β (Baseβπ) β π₯ β (Baseβπ))) | |
27 | ssel 3938 | . . . . . . . 8 β’ ((Baseβπ) β (Baseβπ) β (π¦ β (Baseβπ) β π¦ β (Baseβπ))) | |
28 | 26, 27 | anim12d 610 | . . . . . . 7 β’ ((Baseβπ) β (Baseβπ) β ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯ β (Baseβπ) β§ π¦ β (Baseβπ)))) |
29 | 25, 28 | ax-mp 5 | . . . . . 6 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯ β (Baseβπ) β§ π¦ β (Baseβπ))) |
30 | eqid 2737 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
31 | 1, 30, 18 | efmndov 18692 | . . . . . 6 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) = (π₯ β π¦)) |
32 | 29, 31 | syl 17 | . . . . 5 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) = (π₯ β π¦)) |
33 | 22, 32 | eqtrid 2789 | . . . 4 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) = (π₯ β π¦)) |
34 | 11, 33 | symggrplem 18695 | . . 3 β’ ((π β (Baseβπ) β§ π β (Baseβπ) β§ π β (Baseβπ)) β ((π(+gβπ)π)(+gβπ)π) = (π(+gβπ)(π(+gβπ)π))) |
35 | 34 | rgen3 3200 | . 2 β’ βπ β (Baseβπ)βπ β (Baseβπ)βπ β (Baseβπ)((π(+gβπ)π)(+gβπ)π) = (π(+gβπ)(π(+gβπ)π)) |
36 | 8, 9 | issgrp 18548 | . 2 β’ (π β Smgrp β (π β Mgm β§ βπ β (Baseβπ)βπ β (Baseβπ)βπ β (Baseβπ)((π(+gβπ)π)(+gβπ)π) = (π(+gβπ)(π(+gβπ)π)))) |
37 | 7, 35, 36 | mpbir2an 710 | 1 β’ π β Smgrp |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 Vcvv 3446 βͺ cun 3909 β wss 3911 {csn 4587 βͺ ciun 4955 β¦ cmpt 5189 β ccom 5638 βcfv 6497 (class class class)co 7358 0cc0 11052 βcn 12154 β0cn0 12414 ..^cfzo 13568 mod cmo 13775 Basecbs 17084 βΎs cress 17113 +gcplusg 17134 Mgmcmgm 18496 Smgrpcsgrp 18546 EndoFMndcefmnd 18679 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-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 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fz 13426 df-fzo 13569 df-fl 13698 df-mod 13776 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-tset 17153 df-mgm 18498 df-sgrp 18547 df-efmnd 18680 |
This theorem is referenced by: smndex1mnd 18721 |
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