![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > znle2 | Structured version Visualization version GIF version |
Description: The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znle2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znle2.f | ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) |
znle2.w | ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) |
znle2.l | ⊢ ≤ = (le‘𝑌) |
Ref | Expression |
---|---|
znle2 | ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2692 | . . 3 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
2 | eqid 2692 | . . 3 ⊢ (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
3 | znle2.y | . . 3 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
4 | eqid 2692 | . . 3 ⊢ ((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = ((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) | |
5 | znle2.w | . . 3 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
6 | znle2.l | . . 3 ⊢ ≤ = (le‘𝑌) | |
7 | 1, 2, 3, 4, 5, 6 | znle 19975 | . 2 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) ∘ ≤ ) ∘ ◡((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊))) |
8 | 1, 2, 3 | znzrh 19982 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (ℤRHom‘𝑌)) |
9 | 8 | reseq1d 5470 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = ((ℤRHom‘𝑌) ↾ 𝑊)) |
10 | znle2.f | . . . . 5 ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) | |
11 | 9, 10 | syl6eqr 2744 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = 𝐹) |
12 | 11 | coeq1d 5359 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) ∘ ≤ ) = (𝐹 ∘ ≤ )) |
13 | 11 | cnveqd 5373 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ◡((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = ◡𝐹) |
14 | 12, 13 | coeq12d 5362 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) ∘ ≤ ) ∘ ◡((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊)) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
15 | 7, 14 | eqtrd 2726 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1564 ∈ wcel 2071 ifcif 4162 {csn 4253 ◡ccnv 5185 ↾ cres 5188 ∘ ccom 5190 ‘cfv 5969 (class class class)co 6733 0cc0 10017 ≤ cle 10156 ℕ0cn0 11373 ℤcz 11458 ..^cfzo 12548 lecple 16039 /s cqus 16256 ~QG cqg 17680 RSpancrsp 19262 ℤringzring 19909 ℤRHomczrh 19939 ℤ/nℤczn 19942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-addf 10096 ax-mulf 10097 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-1o 7648 df-oadd 7652 df-er 7830 df-map 7944 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-2 11160 df-3 11161 df-4 11162 df-5 11163 df-6 11164 df-7 11165 df-8 11166 df-9 11167 df-n0 11374 df-z 11459 df-dec 11575 df-uz 11769 df-fz 12409 df-struct 15950 df-ndx 15951 df-slot 15952 df-base 15954 df-sets 15955 df-ress 15956 df-plusg 16045 df-mulr 16046 df-starv 16047 df-tset 16051 df-ple 16052 df-ds 16055 df-unif 16056 df-0g 16193 df-mgm 17332 df-sgrp 17374 df-mnd 17385 df-mhm 17425 df-grp 17515 df-minusg 17516 df-subg 17681 df-ghm 17748 df-cmn 18284 df-mgp 18579 df-ur 18591 df-ring 18638 df-cring 18639 df-rnghom 18806 df-subrg 18869 df-cnfld 19838 df-zring 19910 df-zrh 19943 df-zn 19946 |
This theorem is referenced by: znleval 19994 |
Copyright terms: Public domain | W3C validator |