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Mirrors > Home > MPE Home > Th. List > ablfaclem1 | Structured version Visualization version GIF version |
Description: Lemma for ablfac 19210. (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 2833 | . . . 4 ⊢ (𝑔 = 𝑈 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑠) = 𝑈)) | |
2 | 1 | anbi2d 630 | . . 3 ⊢ (𝑔 = 𝑈 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))) |
3 | 2 | rabbidv 3480 | . 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 6683 | . . . . 5 ⊢ (SubGrp‘𝐺) ∈ V | |
7 | 5, 6 | rabex2 5237 | . . . 4 ⊢ 𝐶 ∈ V |
8 | 7 | wrdexi 13875 | . . 3 ⊢ Word 𝐶 ∈ V |
9 | 8 | rabex 5235 | . 2 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)} ∈ V |
10 | 3, 4, 9 | fvmpt 6768 | 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 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ∩ cin 3935 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ↑cexp 13430 ♯chash 13691 Word cword 13862 ∥ cdvds 15607 ℙcprime 16015 pCnt cpc 16173 Basecbs 16483 ↾s cress 16484 SubGrpcsubg 18273 odcod 18652 pGrp cpgp 18654 Abelcabl 18907 CycGrpccyg 18996 DProd cdprd 19115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-map 8408 df-nn 11639 df-n0 11899 df-word 13863 |
This theorem is referenced by: ablfaclem2 19208 ablfaclem3 19209 ablfac 19210 |
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