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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 12240 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | zntos.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | znhash.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
4 | eqid 2738 | . . . . . 6 ⊢ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) | |
5 | eqid 2738 | . . . . . 6 ⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | 2, 3, 4, 5 | znf1o 20759 | . . . . 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 12007 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
9 | ifnefalse 4471 | . . . . 5 ⊢ (𝑁 ≠ 0 → if(𝑁 = 0, ℤ, (0..^𝑁)) = (0..^𝑁)) | |
10 | f1oeq2 6705 | . . . . 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 7308 | . . . 4 ⊢ (0..^𝑁) ∈ V | |
14 | 13 | f1oen 8761 | . . 3 ⊢ (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵 → (0..^𝑁) ≈ 𝐵) |
15 | ensym 8789 | . . 3 ⊢ ((0..^𝑁) ≈ 𝐵 → 𝐵 ≈ (0..^𝑁)) | |
16 | hasheni 14062 | . . 3 ⊢ (𝐵 ≈ (0..^𝑁) → (♯‘𝐵) = (♯‘(0..^𝑁))) | |
17 | 12, 14, 15, 16 | 4syl 19 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = (♯‘(0..^𝑁))) |
18 | hashfzo0 14145 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁) | |
19 | 1, 18 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁) |
20 | 17, 19 | eqtrd 2778 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ifcif 4459 class class class wbr 5074 ↾ cres 5591 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ≈ cen 8730 0cc0 10871 ℕcn 11973 ℕ0cn0 12233 ℤcz 12319 ..^cfzo 13382 ♯chash 14044 Basecbs 16912 ℤRHomczrh 20701 ℤ/nℤczn 20704 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-hash 14045 df-dvds 15964 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zn 20708 |
This theorem is referenced by: znfi 20767 znfld 20768 znidomb 20769 frlmpwfi 40923 isnumbasgrplem3 40930 cznnring 45514 |
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