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Mirrors > Home > MPE Home > Th. List > cygznlem2a | Structured version Visualization version GIF version |
Description: Lemma for cygzn 20778. (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 6789 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V) | |
3 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
4 | cyggrp 19490 | . . . . . . 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 4015 | . . . . . . 7 ⊢ 𝐸 ⊆ 𝐵 |
10 | cygzn.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
11 | 9, 10 | sselid 3919 | . . . . . 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 18721 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵) |
16 | 6, 7, 12, 15 | syl3anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵) |
17 | fveq2 6774 | . . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘)) | |
18 | oveq1 7282 | . . . 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 20774 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋))) |
23 | 22 | biimpd 228 | . . . . . 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 1348 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋)) |
26 | 1, 2, 16, 17, 18, 25 | fliftfund 7184 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
27 | 1, 2, 16 | fliftf 7186 | . . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)) |
28 | 26, 27 | mpbid 231 | . 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵) |
29 | hashcl 14071 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
30 | 29 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
31 | 0nn0 12248 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
32 | 31 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
33 | 30, 32 | ifclda 4494 | . . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
34 | 19, 33 | eqeltrid 2843 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
35 | eqid 2738 | . . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
36 | 20, 35, 21 | znzrhfo 20755 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌)) |
37 | 34, 36 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌)) |
38 | fof 6688 | . . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) | |
39 | 37, 38 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌)) |
40 | 39 | feqmptd 6837 | . . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
41 | 40 | rneqd 5847 | . . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))) |
42 | forn 6691 | . . . . 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 2780 | . . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌)) |
45 | 44 | feq2d 6586 | . 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵)) |
46 | 28, 45 | mpbid 231 | 1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ifcif 4459 〈cop 4567 ↦ cmpt 5157 ran crn 5590 Fun wfun 6427 ⟶wf 6429 –onto→wfo 6431 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 0cc0 10871 ℕ0cn0 12233 ℤcz 12319 ♯chash 14044 Basecbs 16912 Grpcgrp 18577 .gcmg 18700 CycGrpccyg 19477 ℤRHomczrh 20701 ℤ/nℤczn 20704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-od 19136 df-cmn 19388 df-abl 19389 df-cyg 19478 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zn 20708 |
This theorem is referenced by: cygznlem2 20776 cygznlem3 20777 |
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