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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 2819 | . . 3 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
2 | eqid 2819 | . . 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 2819 | . . 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 20675 | . 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 20681 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (ℤRHom‘𝑌)) |
9 | 8 | reseq1d 5845 | . . . . 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 2872 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = 𝐹) |
12 | 11 | coeq1d 5725 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) ∘ ≤ ) = (𝐹 ∘ ≤ )) |
13 | 11 | cnveqd 5739 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ◡((ℤRHom‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ↾ 𝑊) = ◡𝐹) |
14 | 12, 13 | coeq12d 5728 | . 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 2854 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ifcif 4465 {csn 4559 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7148 0cc0 10529 ≤ cle 10668 ℕ0cn0 11889 ℤcz 11973 ..^cfzo 13025 lecple 16564 /s cqus 16770 ~QG cqg 18267 RSpancrsp 19935 ℤringzring 20609 ℤRHomczrh 20639 ℤ/nℤczn 20642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-subg 18268 df-ghm 18348 df-cmn 18900 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-rnghom 19459 df-subrg 19525 df-cnfld 20538 df-zring 20610 df-zrh 20643 df-zn 20646 |
This theorem is referenced by: znleval 20693 |
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