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Mirrors > Home > MPE Home > Th. List > cygznlem2a | Structured version Visualization version GIF version |
Description: Lemma for cygzn 20382. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
cygzn.b | ⊢ 𝐵 = (Base‘𝐺) |
cygzn.n | ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0) |
cygzn.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
cygzn.m | ⊢ · = (.g‘𝐺) |
cygzn.l | ⊢ 𝐿 = (ℤRHom‘𝑌) |
cygzn.e | ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
cygzn.g | ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
cygzn.x | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
cygzn.f | ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) |
Ref | Expression |
---|---|
cygznlem2a | ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygzn.f | . . . 4 ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) | |
2 | fvexd 6683 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V) | |
3 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
4 | cyggrp 19121 | . . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝐺 ∈ Grp) |
7 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ) | |
8 | cygzn.e | . . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
9 | 8 | ssrab3 3969 | . . . . . . 7 ⊢ 𝐸 ⊆ 𝐵 |
10 | cygzn.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
11 | 9, 10 | sseldi 3873 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑋 ∈ 𝐵) |
13 | cygzn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | cygzn.m | . . . . . 6 ⊢ · = (.g‘𝐺) | |
15 | 13, 14 | mulgcl 18356 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵) |
16 | 6, 7, 12, 15 | syl3anc 1372 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵) |
17 | fveq2 6668 | . . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘)) | |
18 | oveq1 7171 | . . . 4 ⊢ (𝑚 = 𝑘 → (𝑚 · 𝑋) = (𝑘 · 𝑋)) | |
19 | cygzn.n | . . . . . . . 8 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0) | |
20 | cygzn.y | . . . . . . . 8 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
21 | cygzn.l | . . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
22 | 13, 19, 20, 14, 21, 8, 3, 10 | cygznlem1 20378 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋))) |
23 | 22 | biimpd 232 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋))) |
24 | 23 | exp32 424 | . . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℤ → (𝑘 ∈ ℤ → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋))))) |
25 | 24 | 3imp2 1350 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋)) |
26 | 1, 2, 16, 17, 18, 25 | fliftfund 7073 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
27 | 1, 2, 16 | fliftf 7075 | . . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)) |
28 | 26, 27 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵) |
29 | hashcl 13802 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
30 | 29 | adantl 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
31 | 0nn0 11984 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
32 | 31 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
33 | 30, 32 | ifclda 4446 | . . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
34 | 19, 33 | eqeltrid 2837 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
35 | eqid 2738 | . . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
36 | 20, 35, 21 | znzrhfo 20359 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
37 | 34, 36 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌)) |
38 | fof 6586 | . . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) | |
39 | 37, 38 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
40 | 39 | feqmptd 6731 | . . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
41 | 40 | rneqd 5775 | . . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
42 | forn 6589 | . . . . 5 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → ran 𝐿 = (Base‘𝑌)) | |
43 | 37, 42 | syl 17 | . . . 4 ⊢ (𝜑 → ran 𝐿 = (Base‘𝑌)) |
44 | 41, 43 | eqtr3d 2775 | . . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌)) |
45 | 44 | feq2d 6484 | . 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵)) |
46 | 28, 45 | mpbid 235 | 1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 Vcvv 3397 ifcif 4411 〈cop 4519 ↦ cmpt 5107 ran crn 5520 Fun wfun 6327 ⟶wf 6329 –onto→wfo 6331 ‘cfv 6333 (class class class)co 7164 Fincfn 8548 0cc0 10608 ℕ0cn0 11969 ℤcz 12055 ♯chash 13775 Basecbs 16579 Grpcgrp 18212 .gcmg 18335 CycGrpccyg 19108 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-inf2 9170 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-se 5479 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-isom 6342 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-oadd 8128 df-omul 8129 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-oi 9040 df-card 9434 df-acn 9437 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-fl 13246 df-mod 13322 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 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-od 18767 df-cmn 19019 df-abl 19020 df-cyg 19109 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: cygznlem2 20380 cygznlem3 20381 |
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