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Mirrors > Home > MPE Home > Th. List > znleval2 | 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) |
znleval.x | ⊢ 𝑋 = (Base‘𝑌) |
Ref | Expression |
---|---|
znleval2 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znle2.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
2 | znle2.f | . . . 4 ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) | |
3 | znle2.w | . . . 4 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) | |
4 | znle2.l | . . . 4 ⊢ ≤ = (le‘𝑌) | |
5 | znleval.x | . . . 4 ⊢ 𝑋 = (Base‘𝑌) | |
6 | 1, 2, 3, 4, 5 | znleval 20366 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) |
7 | 6 | 3ad2ant1 1134 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) |
8 | 3simpc 1151 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
9 | 8 | biantrurd 536 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) |
10 | df-3an 1090 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) | |
11 | 9, 10 | bitr4di 292 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) |
12 | 7, 11 | bitr4d 285 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ifcif 4411 class class class wbr 5027 ◡ccnv 5518 ↾ cres 5521 ‘cfv 6333 (class class class)co 7164 0cc0 10608 ≤ cle 10747 ℕ0cn0 11969 ℤcz 12055 ..^cfzo 13117 Basecbs 16579 lecple 16668 ℤRHomczrh 20313 ℤ/nℤczn 20316 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
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 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-ec 8315 df-qs 8319 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-rp 12466 df-fz 12975 df-fzo 13118 df-fl 13246 df-mod 13322 df-seq 13454 df-dvds 15693 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-0g 16811 df-imas 16877 df-qus 16878 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mulg 18336 df-subg 18387 df-nsg 18388 df-eqg 18389 df-ghm 18467 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-oppr 19488 df-dvdsr 19506 df-rnghom 19582 df-subrg 19645 df-lmod 19748 df-lss 19816 df-lsp 19856 df-sra 20056 df-rgmod 20057 df-lidl 20058 df-rsp 20059 df-2idl 20117 df-cnfld 20211 df-zring 20283 df-zrh 20317 df-zn 20320 |
This theorem is referenced by: zntoslem 20368 |
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