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Mirrors > Home > MPE Home > Th. List > znle | Structured version Visualization version GIF version |
Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znval.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znval.u | ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
znval.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znval.f | ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) |
znval.w | ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) |
znle.l | ⊢ ≤ = (le‘𝑌) |
Ref | Expression |
---|---|
znle | ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znval.s | . . . 4 ⊢ 𝑆 = (RSpan‘ℤring) | |
2 | znval.u | . . . 4 ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) | |
3 | znval.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
4 | znval.f | . . . 4 ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) | |
5 | znval.w | . . . 4 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | eqid 2651 | . . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | znval 19931 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
8 | 7 | fveq2d 6233 | . 2 ⊢ (𝑁 ∈ ℕ0 → (le‘𝑌) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
9 | znle.l | . 2 ⊢ ≤ = (le‘𝑌) | |
10 | ovex 6718 | . . . 4 ⊢ (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V | |
11 | 2, 10 | eqeltri 2726 | . . 3 ⊢ 𝑈 ∈ V |
12 | fvex 6239 | . . . . . . 7 ⊢ (ℤRHom‘𝑈) ∈ V | |
13 | 12 | resex 5478 | . . . . . 6 ⊢ ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V |
14 | 4, 13 | eqeltri 2726 | . . . . 5 ⊢ 𝐹 ∈ V |
15 | xrex 11867 | . . . . . . 7 ⊢ ℝ* ∈ V | |
16 | 15, 15 | xpex 7004 | . . . . . 6 ⊢ (ℝ* × ℝ*) ∈ V |
17 | lerelxr 10139 | . . . . . 6 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
18 | 16, 17 | ssexi 4836 | . . . . 5 ⊢ ≤ ∈ V |
19 | 14, 18 | coex 7160 | . . . 4 ⊢ (𝐹 ∘ ≤ ) ∈ V |
20 | 14 | cnvex 7155 | . . . 4 ⊢ ◡𝐹 ∈ V |
21 | 19, 20 | coex 7160 | . . 3 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V |
22 | pleid 16096 | . . . 4 ⊢ le = Slot (le‘ndx) | |
23 | 22 | setsid 15961 | . . 3 ⊢ ((𝑈 ∈ V ∧ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
24 | 11, 21, 23 | mp2an 708 | . 2 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
25 | 8, 9, 24 | 3eqtr4g 2710 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ifcif 4119 {csn 4210 〈cop 4216 × cxp 5141 ◡ccnv 5142 ↾ cres 5145 ∘ ccom 5147 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℝ*cxr 10111 ≤ cle 10113 ℕ0cn0 11330 ℤcz 11415 ..^cfzo 12504 ndxcnx 15901 sSet csts 15902 lecple 15995 /s cqus 16212 ~QG cqg 17637 RSpancrsp 19219 ℤringzring 19866 ℤRHomczrh 19896 ℤ/nℤczn 19899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-subg 17638 df-cmn 18241 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-subrg 18826 df-cnfld 19795 df-zring 19867 df-zn 19903 |
This theorem is referenced by: znval2 19933 znle2 19950 |
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