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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem2-rN | Structured version Visualization version GIF version |
Description: Lemma for eringring 38168. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ernggrp.h-r | ⊢ 𝐻 = (LHyp‘𝐾) |
ernggrp.d-r | ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) |
ernggrplem.b-r | ⊢ 𝐵 = (Base‘𝐾) |
ernggrplem.t-r | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
ernggrplem.e-r | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
ernggrplem.p-r | ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) |
ernggrplem.o-r | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
ernggrplem.i-r | ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) |
Ref | Expression |
---|---|
erngdvlem2-rN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ernggrp.h-r | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | ernggrplem.t-r | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | ernggrplem.e-r | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | ernggrp.d-r | . . . 4 ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) | |
5 | eqid 2820 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngbase-rN 37985 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2826 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷)) |
8 | eqid 2820 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
9 | 1, 2, 3, 4, 8 | erngfplus-rN 37986 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) |
10 | ernggrplem.p-r | . . 3 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) | |
11 | 9, 10 | syl6reqr 2874 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | ernggrplem.b-r | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
13 | ernggrplem.o-r | . . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
14 | ernggrplem.i-r | . . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) | |
15 | 1, 4, 12, 2, 3, 10, 13, 14 | erngdvlem1-rN 38172 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
16 | 1, 2, 3, 10 | tendoplcom 37958 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) = (𝑡𝑃𝑠)) |
17 | 7, 11, 15, 16 | isabld 18913 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ↦ cmpt 5139 I cid 5452 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 ‘cfv 6348 ∈ cmpo 7151 Basecbs 16476 +gcplusg 16558 Abelcabl 18900 HLchlt 36526 LHypclh 37160 LTrncltrn 37277 TEndoctendo 37928 EDRingRcedring-rN 37930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-riotaBAD 36129 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 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-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-undef 7932 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-plusg 16571 df-mulr 16572 df-0g 16708 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-p1 17643 df-lat 17649 df-clat 17711 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-cmn 18901 df-abl 18902 df-oposet 36352 df-ol 36354 df-oml 36355 df-covers 36442 df-ats 36443 df-atl 36474 df-cvlat 36498 df-hlat 36527 df-llines 36674 df-lplanes 36675 df-lvols 36676 df-lines 36677 df-psubsp 36679 df-pmap 36680 df-padd 36972 df-lhyp 37164 df-laut 37165 df-ldil 37280 df-ltrn 37281 df-trl 37335 df-tendo 37931 df-edring-rN 37932 |
This theorem is referenced by: (None) |
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