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Mirrors > Home > MPE Home > Th. List > smndex1basss | Structured version Visualization version GIF version |
Description: The modulo function πΌ and the constant functions (πΊβπΎ) are endofunctions on β0. (Contributed by AV, 12-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)}) |
Ref | Expression |
---|---|
smndex1basss | β’ π΅ β (Baseβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1mgm.b | . . . . . 6 β’ π΅ = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) | |
2 | 1 | eleq2i 2817 | . . . . 5 β’ (π β π΅ β π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) |
3 | fveq2 6890 | . . . . . . . . 9 β’ (π = π β (πΊβπ) = (πΊβπ)) | |
4 | 3 | sneqd 4637 | . . . . . . . 8 β’ (π = π β {(πΊβπ)} = {(πΊβπ)}) |
5 | 4 | cbviunv 5039 | . . . . . . 7 β’ βͺ π β (0..^π){(πΊβπ)} = βͺ π β (0..^π){(πΊβπ)} |
6 | 5 | uneq2i 4154 | . . . . . 6 β’ ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) |
7 | 6 | eleq2i 2817 | . . . . 5 β’ (π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) β π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) |
8 | 2, 7 | bitri 274 | . . . 4 β’ (π β π΅ β π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) |
9 | elun 4142 | . . . 4 β’ (π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) β (π β {πΌ} β¨ π β βͺ π β (0..^π){(πΊβπ)})) | |
10 | velsn 4641 | . . . . 5 β’ (π β {πΌ} β π = πΌ) | |
11 | eliun 4996 | . . . . 5 β’ (π β βͺ π β (0..^π){(πΊβπ)} β βπ β (0..^π)π β {(πΊβπ)}) | |
12 | 10, 11 | orbi12i 912 | . . . 4 β’ ((π β {πΌ} β¨ π β βͺ π β (0..^π){(πΊβπ)}) β (π = πΌ β¨ βπ β (0..^π)π β {(πΊβπ)})) |
13 | 8, 9, 12 | 3bitri 296 | . . 3 β’ (π β π΅ β (π = πΌ β¨ βπ β (0..^π)π β {(πΊβπ)})) |
14 | smndex1ibas.m | . . . . . 6 β’ π = (EndoFMndββ0) | |
15 | smndex1ibas.n | . . . . . 6 β’ π β β | |
16 | smndex1ibas.i | . . . . . 6 β’ πΌ = (π₯ β β0 β¦ (π₯ mod π)) | |
17 | 14, 15, 16 | smndex1ibas 18851 | . . . . 5 β’ πΌ β (Baseβπ) |
18 | eleq1 2813 | . . . . 5 β’ (π = πΌ β (π β (Baseβπ) β πΌ β (Baseβπ))) | |
19 | 17, 18 | mpbiri 257 | . . . 4 β’ (π = πΌ β π β (Baseβπ)) |
20 | smndex1ibas.g | . . . . . . . 8 β’ πΊ = (π β (0..^π) β¦ (π₯ β β0 β¦ π)) | |
21 | 14, 15, 16, 20 | smndex1gbas 18853 | . . . . . . 7 β’ (π β (0..^π) β (πΊβπ) β (Baseβπ)) |
22 | 21 | adantr 479 | . . . . . 6 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β (πΊβπ) β (Baseβπ)) |
23 | elsni 4642 | . . . . . . . 8 β’ (π β {(πΊβπ)} β π = (πΊβπ)) | |
24 | 23 | eleq1d 2810 | . . . . . . 7 β’ (π β {(πΊβπ)} β (π β (Baseβπ) β (πΊβπ) β (Baseβπ))) |
25 | 24 | adantl 480 | . . . . . 6 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β (π β (Baseβπ) β (πΊβπ) β (Baseβπ))) |
26 | 22, 25 | mpbird 256 | . . . . 5 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β π β (Baseβπ)) |
27 | 26 | rexlimiva 3137 | . . . 4 β’ (βπ β (0..^π)π β {(πΊβπ)} β π β (Baseβπ)) |
28 | 19, 27 | jaoi 855 | . . 3 β’ ((π = πΌ β¨ βπ β (0..^π)π β {(πΊβπ)}) β π β (Baseβπ)) |
29 | 13, 28 | sylbi 216 | . 2 β’ (π β π΅ β π β (Baseβπ)) |
30 | 29 | ssriv 3977 | 1 β’ π΅ β (Baseβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 β¨ wo 845 = wceq 1533 β wcel 2098 βwrex 3060 βͺ cun 3939 β wss 3941 {csn 4625 βͺ ciun 4992 β¦ cmpt 5227 βcfv 6543 (class class class)co 7413 0cc0 11133 βcn 12237 β0cn0 12497 ..^cfzo 13654 mod cmo 13861 Basecbs 17174 EndoFMndcefmnd 18819 |
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 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-tset 17246 df-efmnd 18820 |
This theorem is referenced by: smndex1bas 18857 smndex1mgm 18858 smndex1sgrp 18859 smndex1mnd 18861 smndex1id 18862 |
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