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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem1-rN | Structured version Visualization version GIF version |
Description: Lemma for eringring 38008. (Contributed by NM, 4-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 |
---|---|
erngdvlem1-rN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
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 2818 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngbase-rN 37825 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2824 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷)) |
8 | eqid 2818 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
9 | 1, 2, 3, 4, 8 | erngfplus-rN 37826 | . . 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 2872 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | 1, 2, 3, 10 | tendoplcl 37797 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸) |
13 | 1, 2, 3, 10 | tendoplass 37799 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑃𝑢) = (𝑠𝑃(𝑡𝑃𝑢))) |
14 | ernggrplem.b-r | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
15 | ernggrplem.o-r | . . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
16 | 14, 1, 2, 3, 15 | tendo0cl 37806 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
17 | 14, 1, 2, 3, 15, 10 | tendo0pl 37807 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑂𝑃𝑠) = 𝑠) |
18 | ernggrplem.i-r | . . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) | |
19 | 1, 2, 3, 18 | tendoicl 37812 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐼‘𝑠) ∈ 𝐸) |
20 | 1, 2, 3, 18, 14, 10, 15 | tendoipl 37813 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ((𝐼‘𝑠)𝑃𝑠) = 𝑂) |
21 | 7, 11, 12, 13, 16, 17, 19, 20 | isgrpd 18063 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 I cid 5452 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 ‘cfv 6348 ∈ cmpo 7147 Basecbs 16471 +gcplusg 16553 Grpcgrp 18041 HLchlt 36366 LHypclh 37000 LTrncltrn 37117 TEndoctendo 37768 EDRingRcedring-rN 37770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tendo 37771 df-edring-rN 37772 |
This theorem is referenced by: erngdvlem2-rN 38013 erngdvlem3-rN 38014 erngdvlem4-rN 38015 |
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