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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 12515 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | zntos.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | znhash.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
4 | eqid 2727 | . . . . . 6 ⊢ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) | |
5 | eqid 2727 | . . . . . 6 ⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | 2, 3, 4, 5 | znf1o 21490 | . . . . 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 12282 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
9 | ifnefalse 4542 | . . . . 5 ⊢ (𝑁 ≠ 0 → if(𝑁 = 0, ℤ, (0..^𝑁)) = (0..^𝑁)) | |
10 | f1oeq2 6831 | . . . . 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 231 | . . 3 ⊢ (𝑁 ∈ ℕ → ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵) |
13 | ovex 7457 | . . . 4 ⊢ (0..^𝑁) ∈ V | |
14 | 13 | f1oen 8998 | . . 3 ⊢ (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵 → (0..^𝑁) ≈ 𝐵) |
15 | ensym 9028 | . . 3 ⊢ ((0..^𝑁) ≈ 𝐵 → 𝐵 ≈ (0..^𝑁)) | |
16 | hasheni 14345 | . . 3 ⊢ (𝐵 ≈ (0..^𝑁) → (♯‘𝐵) = (♯‘(0..^𝑁))) | |
17 | 12, 14, 15, 16 | 4syl 19 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = (♯‘(0..^𝑁))) |
18 | hashfzo0 14427 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁) | |
19 | 1, 18 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁) |
20 | 17, 19 | eqtrd 2767 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ifcif 4530 class class class wbr 5150 ↾ cres 5682 –1-1-onto→wf1o 6550 ‘cfv 6551 (class class class)co 7424 ≈ cen 8965 0cc0 11144 ℕcn 12248 ℕ0cn0 12508 ℤcz 12594 ..^cfzo 13665 ♯chash 14327 Basecbs 17185 ℤRHomczrh 21430 ℤ/nℤczn 21433 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-ec 8731 df-qs 8735 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-hash 14328 df-dvds 16237 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-0g 17428 df-imas 17495 df-qus 17496 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-nsg 19084 df-eqg 19085 df-ghm 19173 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-cring 20181 df-oppr 20278 df-dvdsr 20301 df-rhm 20416 df-subrng 20488 df-subrg 20513 df-lmod 20750 df-lss 20821 df-lsp 20861 df-sra 21063 df-rgmod 21064 df-lidl 21109 df-rsp 21110 df-2idl 21149 df-cnfld 21285 df-zring 21378 df-zrh 21434 df-zn 21437 |
This theorem is referenced by: znfi 21498 znfld 21499 znidomb 21500 frlmpwfi 42525 isnumbasgrplem3 42532 cznnring 47375 |
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