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Mirrors > Home > MPE Home > Th. List > znhash | Structured version Visualization version GIF version |
Description: The ℤ/nℤ structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
zntos.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znhash.1 | ⊢ 𝐵 = (Base‘𝑌) |
Ref | Expression |
---|---|
znhash | ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11626 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | zntos.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | znhash.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
4 | eqid 2825 | . . . . . 6 ⊢ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) | |
5 | eqid 2825 | . . . . . 6 ⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | 2, 3, 4, 5 | znf1o 20259 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):if(𝑁 = 0, ℤ, (0..^𝑁))–1-1-onto→𝐵) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):if(𝑁 = 0, ℤ, (0..^𝑁))–1-1-onto→𝐵) |
8 | nnne0 11386 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
9 | ifnefalse 4318 | . . . . 5 ⊢ (𝑁 ≠ 0 → if(𝑁 = 0, ℤ, (0..^𝑁)) = (0..^𝑁)) | |
10 | f1oeq2 6368 | . . . . 5 ⊢ (if(𝑁 = 0, ℤ, (0..^𝑁)) = (0..^𝑁) → (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):if(𝑁 = 0, ℤ, (0..^𝑁))–1-1-onto→𝐵 ↔ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵)) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):if(𝑁 = 0, ℤ, (0..^𝑁))–1-1-onto→𝐵 ↔ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵)) |
12 | 7, 11 | mpbid 224 | . . 3 ⊢ (𝑁 ∈ ℕ → ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵) |
13 | ovex 6937 | . . . 4 ⊢ (0..^𝑁) ∈ V | |
14 | 13 | f1oen 8243 | . . 3 ⊢ (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵 → (0..^𝑁) ≈ 𝐵) |
15 | ensym 8271 | . . 3 ⊢ ((0..^𝑁) ≈ 𝐵 → 𝐵 ≈ (0..^𝑁)) | |
16 | hasheni 13428 | . . 3 ⊢ (𝐵 ≈ (0..^𝑁) → (♯‘𝐵) = (♯‘(0..^𝑁))) | |
17 | 12, 14, 15, 16 | 4syl 19 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = (♯‘(0..^𝑁))) |
18 | hashfzo0 13506 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁) | |
19 | 1, 18 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁) |
20 | 17, 19 | eqtrd 2861 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ifcif 4306 class class class wbr 4873 ↾ cres 5344 –1-1-onto→wf1o 6122 ‘cfv 6123 (class class class)co 6905 ≈ cen 8219 0cc0 10252 ℕcn 11350 ℕ0cn0 11618 ℤcz 11704 ..^cfzo 12760 ♯chash 13410 Basecbs 16222 ℤRHomczrh 20208 ℤ/nℤczn 20211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-hash 13411 df-dvds 15358 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-0g 16455 df-imas 16521 df-qus 16522 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-sbg 17781 df-mulg 17895 df-subg 17942 df-nsg 17943 df-eqg 17944 df-ghm 18009 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-oppr 18977 df-dvdsr 18995 df-rnghom 19071 df-subrg 19134 df-lmod 19221 df-lss 19289 df-lsp 19331 df-sra 19533 df-rgmod 19534 df-lidl 19535 df-rsp 19536 df-2idl 19593 df-cnfld 20107 df-zring 20179 df-zrh 20212 df-zn 20215 |
This theorem is referenced by: znfi 20267 znfld 20268 znidomb 20269 frlmpwfi 38511 isnumbasgrplem3 38518 cznnring 42803 |
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