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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 2739 | . . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | znval 20366 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
8 | 7 | fveq2d 6690 | . 2 ⊢ (𝑁 ∈ ℕ0 → (le‘𝑌) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
9 | znle.l | . 2 ⊢ ≤ = (le‘𝑌) | |
10 | 2 | ovexi 7216 | . . 3 ⊢ 𝑈 ∈ V |
11 | fvex 6699 | . . . . . . 7 ⊢ (ℤRHom‘𝑈) ∈ V | |
12 | 11 | resex 5883 | . . . . . 6 ⊢ ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V |
13 | 4, 12 | eqeltri 2830 | . . . . 5 ⊢ 𝐹 ∈ V |
14 | xrex 12481 | . . . . . . 7 ⊢ ℝ* ∈ V | |
15 | 14, 14 | xpex 7506 | . . . . . 6 ⊢ (ℝ* × ℝ*) ∈ V |
16 | lerelxr 10794 | . . . . . 6 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
17 | 15, 16 | ssexi 5200 | . . . . 5 ⊢ ≤ ∈ V |
18 | 13, 17 | coex 7673 | . . . 4 ⊢ (𝐹 ∘ ≤ ) ∈ V |
19 | 13 | cnvex 7668 | . . . 4 ⊢ ◡𝐹 ∈ V |
20 | 18, 19 | coex 7673 | . . 3 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V |
21 | pleid 16782 | . . . 4 ⊢ le = Slot (le‘ndx) | |
22 | 21 | setsid 16653 | . . 3 ⊢ ((𝑈 ∈ V ∧ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
23 | 10, 20, 22 | mp2an 692 | . 2 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
24 | 8, 9, 23 | 3eqtr4g 2799 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3400 ifcif 4424 {csn 4526 〈cop 4532 × cxp 5533 ◡ccnv 5534 ↾ cres 5537 ∘ ccom 5539 ‘cfv 6349 (class class class)co 7182 0cc0 10627 ℝ*cxr 10764 ≤ cle 10766 ℕ0cn0 11988 ℤcz 12074 ..^cfzo 13136 ndxcnx 16595 sSet csts 16596 lecple 16687 /s cqus 16893 ~QG cqg 18405 RSpancrsp 20074 ℤringzring 20301 ℤRHomczrh 20332 ℤ/nℤczn 20335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-addf 10706 ax-mulf 10707 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-0g 16830 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-grp 18234 df-minusg 18235 df-subg 18406 df-cmn 19038 df-mgp 19371 df-ur 19383 df-ring 19430 df-cring 19431 df-subrg 19664 df-cnfld 20230 df-zring 20302 df-zn 20339 |
This theorem is referenced by: znval2 20368 znle2 20384 |
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