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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for eringring 39422. (Contributed by NM, 4-Aug-2013.) |
Ref | Expression |
---|---|
ernggrp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ernggrp.d | ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) |
erngdv.b | ⊢ 𝐵 = (Base‘𝐾) |
erngdv.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erngdv.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
erngdv.p | ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) |
erngdv.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
erngdv.i | ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) |
Ref | Expression |
---|---|
erngdvlem1 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ernggrp.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erngdv.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erngdv.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | ernggrp.d | . . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
5 | eqid 2736 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngbase 39231 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2742 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷)) |
8 | erngdv.p | . . 3 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) | |
9 | eqid 2736 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
10 | 1, 2, 3, 4, 9 | erngfplus 39232 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) |
11 | 8, 10 | eqtr4id 2795 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | 1, 2, 3, 8 | tendoplcl 39211 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸) |
13 | 1, 2, 3, 8 | tendoplass 39213 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑃𝑢) = (𝑠𝑃(𝑡𝑃𝑢))) |
14 | erngdv.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
15 | erngdv.o | . . 3 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
16 | 14, 1, 2, 3, 15 | tendo0cl 39220 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸) |
17 | 14, 1, 2, 3, 15, 8 | tendo0pl 39221 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( 0 𝑃𝑠) = 𝑠) |
18 | erngdv.i | . . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) | |
19 | 1, 2, 3, 18 | tendoicl 39226 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐼‘𝑠) ∈ 𝐸) |
20 | 1, 2, 3, 18, 14, 8, 15 | tendoipl 39227 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ((𝐼‘𝑠)𝑃𝑠) = 0 ) |
21 | 7, 11, 12, 13, 16, 17, 19, 20 | isgrpd 18764 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5186 I cid 5528 ◡ccnv 5630 ↾ cres 5633 ∘ ccom 5635 ‘cfv 6493 ∈ cmpo 7355 Basecbs 17075 +gcplusg 17125 Grpcgrp 18740 HLchlt 37779 LHypclh 38414 LTrncltrn 38531 TEndoctendo 39182 EDRingcedring 39183 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-riotaBAD 37382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-undef 8200 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-slot 17046 df-ndx 17058 df-base 17076 df-plusg 17138 df-mulr 17139 df-0g 17315 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-p1 18307 df-lat 18313 df-clat 18380 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 df-lvols 37930 df-lines 37931 df-psubsp 37933 df-pmap 37934 df-padd 38226 df-lhyp 38418 df-laut 38419 df-ldil 38534 df-ltrn 38535 df-trl 38589 df-tendo 39185 df-edring 39187 |
This theorem is referenced by: erngdvlem2N 39419 erngdvlem3 39420 erngdvlem4 39421 |
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