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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 2824 | . . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | znval 20685 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
8 | 7 | fveq2d 6677 | . 2 ⊢ (𝑁 ∈ ℕ0 → (le‘𝑌) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
9 | znle.l | . 2 ⊢ ≤ = (le‘𝑌) | |
10 | 2 | ovexi 7193 | . . 3 ⊢ 𝑈 ∈ V |
11 | fvex 6686 | . . . . . . 7 ⊢ (ℤRHom‘𝑈) ∈ V | |
12 | 11 | resex 5902 | . . . . . 6 ⊢ ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V |
13 | 4, 12 | eqeltri 2912 | . . . . 5 ⊢ 𝐹 ∈ V |
14 | xrex 12389 | . . . . . . 7 ⊢ ℝ* ∈ V | |
15 | 14, 14 | xpex 7479 | . . . . . 6 ⊢ (ℝ* × ℝ*) ∈ V |
16 | lerelxr 10707 | . . . . . 6 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
17 | 15, 16 | ssexi 5229 | . . . . 5 ⊢ ≤ ∈ V |
18 | 13, 17 | coex 7638 | . . . 4 ⊢ (𝐹 ∘ ≤ ) ∈ V |
19 | 13 | cnvex 7633 | . . . 4 ⊢ ◡𝐹 ∈ V |
20 | 18, 19 | coex 7638 | . . 3 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V |
21 | pleid 16670 | . . . 4 ⊢ le = Slot (le‘ndx) | |
22 | 21 | setsid 16541 | . . 3 ⊢ ((𝑈 ∈ V ∧ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
23 | 10, 20, 22 | mp2an 690 | . 2 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
24 | 8, 9, 23 | 3eqtr4g 2884 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ifcif 4470 {csn 4570 〈cop 4576 × cxp 5556 ◡ccnv 5557 ↾ cres 5560 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 0cc0 10540 ℝ*cxr 10677 ≤ cle 10679 ℕ0cn0 11900 ℤcz 11984 ..^cfzo 13036 ndxcnx 16483 sSet csts 16484 lecple 16575 /s cqus 16781 ~QG cqg 18278 RSpancrsp 19946 ℤringzring 20620 ℤRHomczrh 20650 ℤ/nℤczn 20653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-subg 18279 df-cmn 18911 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-subrg 19536 df-cnfld 20549 df-zring 20621 df-zn 20657 |
This theorem is referenced by: znval2 20687 znle2 20703 |
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