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Mirrors > Home > MPE Home > Th. List > smndex1id | Structured version Visualization version GIF version |
Description: The modulo function πΌ is the identity of the monoid of endofunctions on β0 restricted to the modulo function πΌ and the constant functions (πΊβπΎ). (Contributed by AV, 16-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 |
---|---|
smndex1id | β’ πΌ = (0gβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.i | . . . . . 6 β’ πΌ = (π₯ β β0 β¦ (π₯ mod π)) | |
2 | nn0ex 12420 | . . . . . . 7 β’ β0 β V | |
3 | 2 | mptex 7174 | . . . . . 6 β’ (π₯ β β0 β¦ (π₯ mod π)) β V |
4 | 1, 3 | eqeltri 2834 | . . . . 5 β’ πΌ β V |
5 | 4 | snid 4623 | . . . 4 β’ πΌ β {πΌ} |
6 | elun1 4137 | . . . 4 β’ (πΌ β {πΌ} β πΌ β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) | |
7 | 5, 6 | ax-mp 5 | . . 3 β’ πΌ β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) |
8 | smndex1mgm.b | . . 3 β’ π΅ = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) | |
9 | 7, 8 | eleqtrri 2837 | . 2 β’ πΌ β π΅ |
10 | smndex1ibas.m | . . . . . 6 β’ π = (EndoFMndββ0) | |
11 | smndex1ibas.n | . . . . . 6 β’ π β β | |
12 | smndex1ibas.g | . . . . . 6 β’ πΊ = (π β (0..^π) β¦ (π₯ β β0 β¦ π)) | |
13 | smndex1mgm.s | . . . . . 6 β’ π = (π βΎs π΅) | |
14 | 10, 11, 1, 12, 8, 13 | smndex1bas 18717 | . . . . 5 β’ (Baseβπ) = π΅ |
15 | 14 | eqcomi 2746 | . . . 4 β’ π΅ = (Baseβπ) |
16 | 15 | a1i 11 | . . 3 β’ (πΌ β π΅ β π΅ = (Baseβπ)) |
17 | snex 5389 | . . . . . 6 β’ {πΌ} β V | |
18 | ovex 7391 | . . . . . . 7 β’ (0..^π) β V | |
19 | snex 5389 | . . . . . . 7 β’ {(πΊβπ)} β V | |
20 | 18, 19 | iunex 7902 | . . . . . 6 β’ βͺ π β (0..^π){(πΊβπ)} β V |
21 | 17, 20 | unex 7681 | . . . . 5 β’ ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) β V |
22 | 8, 21 | eqeltri 2834 | . . . 4 β’ π΅ β V |
23 | eqid 2737 | . . . . 5 β’ (+gβπ) = (+gβπ) | |
24 | 13, 23 | ressplusg 17172 | . . . 4 β’ (π΅ β V β (+gβπ) = (+gβπ)) |
25 | 22, 24 | mp1i 13 | . . 3 β’ (πΌ β π΅ β (+gβπ) = (+gβπ)) |
26 | id 22 | . . 3 β’ (πΌ β π΅ β πΌ β π΅) | |
27 | 10, 11, 1 | smndex1ibas 18711 | . . . . . 6 β’ πΌ β (Baseβπ) |
28 | 27 | a1i 11 | . . . . 5 β’ (πΌ β π΅ β πΌ β (Baseβπ)) |
29 | 10, 11, 1, 12, 8 | smndex1basss 18716 | . . . . . 6 β’ π΅ β (Baseβπ) |
30 | 29 | sseli 3941 | . . . . 5 β’ (π β π΅ β π β (Baseβπ)) |
31 | eqid 2737 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
32 | 10, 31, 23 | efmndov 18692 | . . . . 5 β’ ((πΌ β (Baseβπ) β§ π β (Baseβπ)) β (πΌ(+gβπ)π) = (πΌ β π)) |
33 | 28, 30, 32 | syl2an 597 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ(+gβπ)π) = (πΌ β π)) |
34 | 10, 11, 1, 12, 8, 13 | smndex1mndlem 18720 | . . . . . 6 β’ (π β π΅ β ((πΌ β π) = π β§ (π β πΌ) = π)) |
35 | 34 | simpld 496 | . . . . 5 β’ (π β π΅ β (πΌ β π) = π) |
36 | 35 | adantl 483 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ β π) = π) |
37 | 33, 36 | eqtrd 2777 | . . 3 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ(+gβπ)π) = π) |
38 | 10, 31, 23 | efmndov 18692 | . . . . 5 β’ ((π β (Baseβπ) β§ πΌ β (Baseβπ)) β (π(+gβπ)πΌ) = (π β πΌ)) |
39 | 30, 28, 38 | syl2anr 598 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (π(+gβπ)πΌ) = (π β πΌ)) |
40 | 34 | simprd 497 | . . . . 5 β’ (π β π΅ β (π β πΌ) = π) |
41 | 40 | adantl 483 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (π β πΌ) = π) |
42 | 39, 41 | eqtrd 2777 | . . 3 β’ ((πΌ β π΅ β§ π β π΅) β (π(+gβπ)πΌ) = π) |
43 | 16, 25, 26, 37, 42 | grpidd 18527 | . 2 β’ (πΌ β π΅ β πΌ = (0gβπ)) |
44 | 9, 43 | ax-mp 5 | 1 β’ πΌ = (0gβπ) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3446 βͺ cun 3909 {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 0gc0g 17322 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-0g 17324 df-efmnd 18680 |
This theorem is referenced by: (None) |
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