Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cygznlem2a | Structured version Visualization version GIF version |
Description: Lemma for cygzn 20645. (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 6678 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V) | |
3 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
4 | cyggrp 18938 | . . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝐺 ∈ Grp) |
7 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ) | |
8 | cygzn.e | . . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
9 | 8 | ssrab3 4054 | . . . . . . 7 ⊢ 𝐸 ⊆ 𝐵 |
10 | cygzn.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
11 | 9, 10 | sseldi 3962 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑋 ∈ 𝐵) |
13 | cygzn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | cygzn.m | . . . . . 6 ⊢ · = (.g‘𝐺) | |
15 | 13, 14 | mulgcl 18183 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵) |
16 | 6, 7, 12, 15 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵) |
17 | fveq2 6663 | . . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘)) | |
18 | oveq1 7152 | . . . 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 20641 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋))) |
23 | 22 | biimpd 230 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋))) |
24 | 23 | exp32 421 | . . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℤ → (𝑘 ∈ ℤ → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋))))) |
25 | 24 | 3imp2 1341 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋)) |
26 | 1, 2, 16, 17, 18, 25 | fliftfund 7055 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
27 | 1, 2, 16 | fliftf 7057 | . . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)) |
28 | 26, 27 | mpbid 233 | . 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵) |
29 | hashcl 13705 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
30 | 29 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
31 | 0nn0 11900 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
32 | 31 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
33 | 30, 32 | ifclda 4497 | . . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
34 | 19, 33 | eqeltrid 2914 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
35 | eqid 2818 | . . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
36 | 20, 35, 21 | znzrhfo 20622 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
37 | 34, 36 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌)) |
38 | fof 6583 | . . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) | |
39 | 37, 38 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
40 | 39 | feqmptd 6726 | . . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
41 | 40 | rneqd 5801 | . . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
42 | forn 6586 | . . . . 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 2855 | . . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌)) |
45 | 44 | feq2d 6493 | . 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵)) |
46 | 28, 45 | mpbid 233 | 1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ifcif 4463 〈cop 4563 ↦ cmpt 5137 ran crn 5549 Fun wfun 6342 ⟶wf 6344 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 0cc0 10525 ℕ0cn0 11885 ℤcz 11969 ♯chash 13678 Basecbs 16471 Grpcgrp 18041 .gcmg 18162 CycGrpccyg 18925 ℤRHomczrh 20575 ℤ/nℤczn 20578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-od 18585 df-cmn 18837 df-abl 18838 df-cyg 18926 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zn 20582 |
This theorem is referenced by: cygznlem2 20643 cygznlem3 20644 |
Copyright terms: Public domain | W3C validator |