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Mirrors > Home > MPE Home > Th. List > ablfaclem1 | Structured version Visualization version GIF version |
Description: Lemma for ablfac 19203. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
Ref | Expression |
---|---|
ablfac.b | ⊢ 𝐵 = (Base‘𝐺) |
ablfac.c | ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} |
ablfac.1 | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablfac.2 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
ablfac.o | ⊢ 𝑂 = (od‘𝐺) |
ablfac.a | ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} |
ablfac.s | ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) |
ablfac.w | ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) |
Ref | Expression |
---|---|
ablfaclem1 | ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊‘𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2810 | . . . 4 ⊢ (𝑔 = 𝑈 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑠) = 𝑈)) | |
2 | 1 | anbi2d 631 | . . 3 ⊢ (𝑔 = 𝑈 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))) |
3 | 2 | rabbidv 3427 | . 2 ⊢ (𝑔 = 𝑈 → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)}) |
4 | ablfac.w | . 2 ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) | |
5 | ablfac.c | . . . . 5 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} | |
6 | fvex 6658 | . . . . 5 ⊢ (SubGrp‘𝐺) ∈ V | |
7 | 5, 6 | rabex2 5201 | . . . 4 ⊢ 𝐶 ∈ V |
8 | 7 | wrdexi 13869 | . . 3 ⊢ Word 𝐶 ∈ V |
9 | 8 | rabex 5199 | . 2 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)} ∈ V |
10 | 3, 4, 9 | fvmpt 6745 | 1 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊‘𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ↑cexp 13425 ♯chash 13686 Word cword 13857 ∥ cdvds 15599 ℙcprime 16005 pCnt cpc 16163 Basecbs 16475 ↾s cress 16476 SubGrpcsubg 18265 odcod 18644 pGrp cpgp 18646 Abelcabl 18899 CycGrpccyg 18989 DProd cdprd 19108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-nn 11626 df-n0 11886 df-word 13858 |
This theorem is referenced by: ablfaclem2 19201 ablfaclem3 19202 ablfac 19203 |
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