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Mirrors > Home > MPE Home > Th. List > smndex1iidm | Structured version Visualization version GIF version |
Description: The modulo function πΌ is idempotent. (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)) |
Ref | Expression |
---|---|
smndex1iidm | β’ (πΌ β πΌ) = πΌ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12423 | . . . . 5 β’ (π¦ β β0 β π¦ β β) | |
2 | smndex1ibas.n | . . . . . 6 β’ π β β | |
3 | nnrp 12927 | . . . . . 6 β’ (π β β β π β β+) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 β’ π β β+ |
5 | modabs2 13811 | . . . . 5 β’ ((π¦ β β β§ π β β+) β ((π¦ mod π) mod π) = (π¦ mod π)) | |
6 | 1, 4, 5 | sylancl 587 | . . . 4 β’ (π¦ β β0 β ((π¦ mod π) mod π) = (π¦ mod π)) |
7 | 6 | eqcomd 2743 | . . 3 β’ (π¦ β β0 β (π¦ mod π) = ((π¦ mod π) mod π)) |
8 | 7 | mpteq2ia 5209 | . 2 β’ (π¦ β β0 β¦ (π¦ mod π)) = (π¦ β β0 β¦ ((π¦ mod π) mod π)) |
9 | smndex1ibas.i | . . 3 β’ πΌ = (π₯ β β0 β¦ (π₯ mod π)) | |
10 | oveq1 7365 | . . . 4 β’ (π₯ = π¦ β (π₯ mod π) = (π¦ mod π)) | |
11 | 10 | cbvmptv 5219 | . . 3 β’ (π₯ β β0 β¦ (π₯ mod π)) = (π¦ β β0 β¦ (π¦ mod π)) |
12 | 9, 11 | eqtri 2765 | . 2 β’ πΌ = (π¦ β β0 β¦ (π¦ mod π)) |
13 | nn0z 12525 | . . . . . . 7 β’ (π¦ β β0 β π¦ β β€) | |
14 | 13 | anim2i 618 | . . . . . 6 β’ ((π β β β§ π¦ β β0) β (π β β β§ π¦ β β€)) |
15 | 14 | ancomd 463 | . . . . 5 β’ ((π β β β§ π¦ β β0) β (π¦ β β€ β§ π β β)) |
16 | zmodcl 13797 | . . . . 5 β’ ((π¦ β β€ β§ π β β) β (π¦ mod π) β β0) | |
17 | 15, 16 | syl 17 | . . . 4 β’ ((π β β β§ π¦ β β0) β (π¦ mod π) β β0) |
18 | 12 | a1i 11 | . . . 4 β’ (π β β β πΌ = (π¦ β β0 β¦ (π¦ mod π))) |
19 | 9 | a1i 11 | . . . 4 β’ (π β β β πΌ = (π₯ β β0 β¦ (π₯ mod π))) |
20 | oveq1 7365 | . . . 4 β’ (π₯ = (π¦ mod π) β (π₯ mod π) = ((π¦ mod π) mod π)) | |
21 | 17, 18, 19, 20 | fmptco 7076 | . . 3 β’ (π β β β (πΌ β πΌ) = (π¦ β β0 β¦ ((π¦ mod π) mod π))) |
22 | 2, 21 | ax-mp 5 | . 2 β’ (πΌ β πΌ) = (π¦ β β0 β¦ ((π¦ mod π) mod π)) |
23 | 8, 12, 22 | 3eqtr4ri 2776 | 1 β’ (πΌ β πΌ) = πΌ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5189 β ccom 5638 βcfv 6497 (class class class)co 7358 βcr 11051 βcn 12154 β0cn0 12414 β€cz 12500 β+crp 12916 mod cmo 13775 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-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-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-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 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-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fl 13698 df-mod 13776 |
This theorem is referenced by: smndex1mgm 18718 smndex1mndlem 18720 |
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