![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ablfaclem1 | Structured version Visualization version GIF version |
Description: Lemma for ablfac 20047. (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 2737 | . . . 4 ⊢ (𝑔 = 𝑈 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑠) = 𝑈)) | |
2 | 1 | anbi2d 628 | . . 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 6904 | . . . . 5 ⊢ (SubGrp‘𝐺) ∈ V | |
7 | 5, 6 | rabex2 5331 | . . . 4 ⊢ 𝐶 ∈ V |
8 | 7 | wrdexi 14506 | . . 3 ⊢ Word 𝐶 ∈ V |
9 | 8 | rabex 5329 | . 2 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)} ∈ V |
10 | 3, 4, 9 | fvmpt 6999 | 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 394 = wceq 1533 ∈ wcel 2098 {crab 3419 ∩ cin 3939 class class class wbr 5143 ↦ cmpt 5226 dom cdm 5672 ran crn 5673 ‘cfv 6542 (class class class)co 7415 Fincfn 8960 ↑cexp 14056 ♯chash 14319 Word cword 14494 ∥ cdvds 16228 ℙcprime 16639 pCnt cpc 16802 Basecbs 17177 ↾s cress 17206 SubGrpcsubg 19077 odcod 19481 pGrp cpgp 19483 Abelcabl 19738 CycGrpccyg 19834 DProd cdprd 19952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-1cn 11194 ax-addcl 11196 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-map 8843 df-nn 12241 df-n0 12501 df-word 14495 |
This theorem is referenced by: ablfaclem2 20045 ablfaclem3 20046 ablfac 20047 |
Copyright terms: Public domain | W3C validator |