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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for eringring 40691. (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 2726 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngbase 40500 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2732 | . 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 2726 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
10 | 1, 2, 3, 4, 9 | erngfplus 40501 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) |
11 | 8, 10 | eqtr4id 2785 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | 1, 2, 3, 8 | tendoplcl 40480 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸) |
13 | 1, 2, 3, 8 | tendoplass 40482 | . 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 40489 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸) |
17 | 14, 1, 2, 3, 15, 8 | tendo0pl 40490 | . 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 40495 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐼‘𝑠) ∈ 𝐸) |
20 | 1, 2, 3, 18, 14, 8, 15 | tendoipl 40496 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ((𝐼‘𝑠)𝑃𝑠) = 0 ) |
21 | 7, 11, 12, 13, 16, 17, 19, 20 | isgrpd 18953 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5236 I cid 5579 ◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 ‘cfv 6554 ∈ cmpo 7426 Basecbs 17213 +gcplusg 17266 Grpcgrp 18928 HLchlt 39048 LHypclh 39683 LTrncltrn 39800 TEndoctendo 40451 EDRingcedring 40452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-0g 17456 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tendo 40454 df-edring 40456 |
This theorem is referenced by: erngdvlem2N 40688 erngdvlem3 40689 erngdvlem4 40690 |
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