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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 2820 | . . . . 5 β’ (π β π΅ β π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) |
3 | fveq2 6891 | . . . . . . . . 9 β’ (π = π β (πΊβπ) = (πΊβπ)) | |
4 | 3 | sneqd 4636 | . . . . . . . 8 β’ (π = π β {(πΊβπ)} = {(πΊβπ)}) |
5 | 4 | cbviunv 5037 | . . . . . . 7 β’ βͺ π β (0..^π){(πΊβπ)} = βͺ π β (0..^π){(πΊβπ)} |
6 | 5 | uneq2i 4156 | . . . . . 6 β’ ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) = ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)}) |
7 | 6 | eleq2i 2820 | . . . . 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 275 | . . . 4 β’ (π β π΅ β π β ({πΌ} βͺ βͺ π β (0..^π){(πΊβπ)})) |
9 | elun 4144 | . . . 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 4640 | . . . . 5 β’ (π β {πΌ} β π = πΌ) | |
11 | eliun 4995 | . . . . 5 β’ (π β βͺ π β (0..^π){(πΊβπ)} β βπ β (0..^π)π β {(πΊβπ)}) | |
12 | 10, 11 | orbi12i 913 | . . . 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 297 | . . 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 18837 | . . . . 5 β’ πΌ β (Baseβπ) |
18 | eleq1 2816 | . . . . 5 β’ (π = πΌ β (π β (Baseβπ) β πΌ β (Baseβπ))) | |
19 | 17, 18 | mpbiri 258 | . . . 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 18839 | . . . . . . 7 β’ (π β (0..^π) β (πΊβπ) β (Baseβπ)) |
22 | 21 | adantr 480 | . . . . . 6 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β (πΊβπ) β (Baseβπ)) |
23 | elsni 4641 | . . . . . . . 8 β’ (π β {(πΊβπ)} β π = (πΊβπ)) | |
24 | 23 | eleq1d 2813 | . . . . . . 7 β’ (π β {(πΊβπ)} β (π β (Baseβπ) β (πΊβπ) β (Baseβπ))) |
25 | 24 | adantl 481 | . . . . . 6 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β (π β (Baseβπ) β (πΊβπ) β (Baseβπ))) |
26 | 22, 25 | mpbird 257 | . . . . 5 β’ ((π β (0..^π) β§ π β {(πΊβπ)}) β π β (Baseβπ)) |
27 | 26 | rexlimiva 3142 | . . . 4 β’ (βπ β (0..^π)π β {(πΊβπ)} β π β (Baseβπ)) |
28 | 19, 27 | jaoi 856 | . . 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 3982 | 1 β’ π΅ β (Baseβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 β¨ wo 846 = wceq 1534 β wcel 2099 βwrex 3065 βͺ cun 3942 β wss 3944 {csn 4624 βͺ ciun 4991 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 0cc0 11124 βcn 12228 β0cn0 12488 ..^cfzo 13645 mod cmo 13852 Basecbs 17165 EndoFMndcefmnd 18805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-struct 17101 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-tset 17237 df-efmnd 18806 |
This theorem is referenced by: smndex1bas 18843 smndex1mgm 18844 smndex1sgrp 18845 smndex1mnd 18847 smndex1id 18848 |
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