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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 12480 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | zntos.y | . . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
3 | znhash.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
4 | eqid 2726 | . . . . . 6 ⊢ ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) | |
5 | eqid 2726 | . . . . . 6 ⊢ if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | 2, 3, 4, 5 | znf1o 21442 | . . . . 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 12247 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
9 | ifnefalse 4535 | . . . . 5 ⊢ (𝑁 ≠ 0 → if(𝑁 = 0, ℤ, (0..^𝑁)) = (0..^𝑁)) | |
10 | f1oeq2 6815 | . . . . 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 7437 | . . . 4 ⊢ (0..^𝑁) ∈ V | |
14 | 13 | f1oen 8968 | . . 3 ⊢ (((ℤRHom‘𝑌) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))):(0..^𝑁)–1-1-onto→𝐵 → (0..^𝑁) ≈ 𝐵) |
15 | ensym 8998 | . . 3 ⊢ ((0..^𝑁) ≈ 𝐵 → 𝐵 ≈ (0..^𝑁)) | |
16 | hasheni 14311 | . . 3 ⊢ (𝐵 ≈ (0..^𝑁) → (♯‘𝐵) = (♯‘(0..^𝑁))) | |
17 | 12, 14, 15, 16 | 4syl 19 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = (♯‘(0..^𝑁))) |
18 | hashfzo0 14393 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁) | |
19 | 1, 18 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁) |
20 | 17, 19 | eqtrd 2766 | 1 ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ifcif 4523 class class class wbr 5141 ↾ cres 5671 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 ≈ cen 8935 0cc0 11109 ℕcn 12213 ℕ0cn0 12473 ℤcz 12559 ..^cfzo 13630 ♯chash 14293 Basecbs 17151 ℤRHomczrh 21382 ℤ/nℤczn 21385 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-ec 8704 df-qs 8708 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-hash 14294 df-dvds 16203 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-imas 17461 df-qus 17462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-nsg 19049 df-eqg 19050 df-ghm 19137 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-rhm 20372 df-subrng 20444 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-sra 21019 df-rgmod 21020 df-lidl 21065 df-rsp 21066 df-2idl 21105 df-cnfld 21237 df-zring 21330 df-zrh 21386 df-zn 21389 |
This theorem is referenced by: znfi 21450 znfld 21451 znidomb 21452 frlmpwfi 42399 isnumbasgrplem3 42406 cznnring 47193 |
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