![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2735 | . . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | znval 21568 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉)) |
8 | 7 | fveq2d 6911 | . 2 ⊢ (𝑁 ∈ ℕ0 → (le‘𝑌) = (le‘(𝑈 sSet 〈(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ◡𝐹)〉))) |
9 | znle.l | . 2 ⊢ ≤ = (le‘𝑌) | |
10 | 2 | ovexi 7465 | . . 3 ⊢ 𝑈 ∈ V |
11 | fvex 6920 | . . . . . . 7 ⊢ (ℤRHom‘𝑈) ∈ V | |
12 | 11 | resex 6049 | . . . . . 6 ⊢ ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V |
13 | 4, 12 | eqeltri 2835 | . . . . 5 ⊢ 𝐹 ∈ V |
14 | xrex 13027 | . . . . . . 7 ⊢ ℝ* ∈ V | |
15 | 14, 14 | xpex 7772 | . . . . . 6 ⊢ (ℝ* × ℝ*) ∈ V |
16 | lerelxr 11322 | . . . . . 6 ⊢ ≤ ⊆ (ℝ* × ℝ*) | |
17 | 15, 16 | ssexi 5328 | . . . . 5 ⊢ ≤ ∈ V |
18 | 13, 17 | coex 7953 | . . . 4 ⊢ (𝐹 ∘ ≤ ) ∈ V |
19 | 13 | cnvex 7948 | . . . 4 ⊢ ◡𝐹 ∈ V |
20 | 18, 19 | coex 7953 | . . 3 ⊢ ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V |
21 | pleid 17413 | . . . 4 ⊢ le = Slot (le‘ndx) | |
22 | 21 | setsid 17242 | . . 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 2800 | 1 ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ifcif 4531 {csn 4631 〈cop 4637 × cxp 5687 ◡ccnv 5688 ↾ cres 5691 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℝ*cxr 11292 ≤ cle 11294 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 sSet csts 17197 ndxcnx 17227 lecple 17305 /s cqus 17552 ~QG cqg 19153 RSpancrsp 21235 ℤringczring 21475 ℤRHomczrh 21528 ℤ/nℤczn 21531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-cnfld 21383 df-zring 21476 df-zn 21535 |
This theorem is referenced by: znval2 21570 znle2 21590 |
Copyright terms: Public domain | W3C validator |