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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 12482 | . . . . . . 7 β’ β0 β V | |
3 | 2 | mptex 7220 | . . . . . 6 β’ (π₯ β β0 β¦ (π₯ mod π)) β V |
4 | 1, 3 | eqeltri 2823 | . . . . 5 β’ πΌ β V |
5 | 4 | snid 4659 | . . . 4 β’ πΌ β {πΌ} |
6 | elun1 4171 | . . . 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 2826 | . 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 18831 | . . . . 5 β’ (Baseβπ) = π΅ |
15 | 14 | eqcomi 2735 | . . . 4 β’ π΅ = (Baseβπ) |
16 | 15 | a1i 11 | . . 3 β’ (πΌ β π΅ β π΅ = (Baseβπ)) |
17 | snex 5424 | . . . . . 6 β’ {πΌ} β V | |
18 | ovex 7438 | . . . . . . 7 β’ (0..^π) β V | |
19 | snex 5424 | . . . . . . 7 β’ {(πΊβπ)} β V | |
20 | 18, 19 | iunex 7954 | . . . . . 6 β’ βͺ π β (0..^π){(πΊβπ)} β V |
21 | 17, 20 | unex 7730 | . . . . 5 β’ ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) β V |
22 | 8, 21 | eqeltri 2823 | . . . 4 β’ π΅ β V |
23 | eqid 2726 | . . . . 5 β’ (+gβπ) = (+gβπ) | |
24 | 13, 23 | ressplusg 17244 | . . . 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 18825 | . . . . . 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 18830 | . . . . . 6 β’ π΅ β (Baseβπ) |
30 | 29 | sseli 3973 | . . . . 5 β’ (π β π΅ β π β (Baseβπ)) |
31 | eqid 2726 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
32 | 10, 31, 23 | efmndov 18806 | . . . . 5 β’ ((πΌ β (Baseβπ) β§ π β (Baseβπ)) β (πΌ(+gβπ)π) = (πΌ β π)) |
33 | 28, 30, 32 | syl2an 595 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ(+gβπ)π) = (πΌ β π)) |
34 | 10, 11, 1, 12, 8, 13 | smndex1mndlem 18834 | . . . . . 6 β’ (π β π΅ β ((πΌ β π) = π β§ (π β πΌ) = π)) |
35 | 34 | simpld 494 | . . . . 5 β’ (π β π΅ β (πΌ β π) = π) |
36 | 35 | adantl 481 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ β π) = π) |
37 | 33, 36 | eqtrd 2766 | . . 3 β’ ((πΌ β π΅ β§ π β π΅) β (πΌ(+gβπ)π) = π) |
38 | 10, 31, 23 | efmndov 18806 | . . . . 5 β’ ((π β (Baseβπ) β§ πΌ β (Baseβπ)) β (π(+gβπ)πΌ) = (π β πΌ)) |
39 | 30, 28, 38 | syl2anr 596 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (π(+gβπ)πΌ) = (π β πΌ)) |
40 | 34 | simprd 495 | . . . . 5 β’ (π β π΅ β (π β πΌ) = π) |
41 | 40 | adantl 481 | . . . 4 β’ ((πΌ β π΅ β§ π β π΅) β (π β πΌ) = π) |
42 | 39, 41 | eqtrd 2766 | . . 3 β’ ((πΌ β π΅ β§ π β π΅) β (π(+gβπ)πΌ) = π) |
43 | 16, 25, 26, 37, 42 | grpidd 18604 | . 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 βͺ cun 3941 {csn 4623 βͺ ciun 4990 β¦ cmpt 5224 β ccom 5673 βcfv 6537 (class class class)co 7405 0cc0 11112 βcn 12216 β0cn0 12476 ..^cfzo 13633 mod cmo 13840 Basecbs 17153 βΎs cress 17182 +gcplusg 17206 0gc0g 17394 EndoFMndcefmnd 18793 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-tset 17225 df-0g 17396 df-efmnd 18794 |
This theorem is referenced by: (None) |
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