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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem2N | Structured version Visualization version GIF version |
Description: Lemma for eringring 39666. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.) |
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 |
---|---|
erngdvlem2N | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Abel) |
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 2731 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngbase 39475 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2737 | . 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 2731 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
10 | 1, 2, 3, 4, 9 | erngfplus 39476 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) |
11 | 8, 10 | eqtr4id 2790 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | erngdv.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
13 | erngdv.o | . . 3 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
14 | erngdv.i | . . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) | |
15 | 1, 4, 12, 2, 3, 8, 13, 14 | erngdvlem1 39662 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
16 | 1, 2, 3, 8 | tendoplcom 39456 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) = (𝑡𝑃𝑠)) |
17 | 7, 11, 15, 16 | isabld 19627 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5224 I cid 5566 ◡ccnv 5668 ↾ cres 5671 ∘ ccom 5673 ‘cfv 6532 ∈ cmpo 7395 Basecbs 17126 +gcplusg 17179 Abelcabl 19613 HLchlt 38023 LHypclh 38658 LTrncltrn 38775 TEndoctendo 39426 EDRingcedring 39427 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-riotaBAD 37626 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-undef 8240 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-mulr 17193 df-0g 17369 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-cmn 19614 df-abl 19615 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 df-llines 38172 df-lplanes 38173 df-lvols 38174 df-lines 38175 df-psubsp 38177 df-pmap 38178 df-padd 38470 df-lhyp 38662 df-laut 38663 df-ldil 38778 df-ltrn 38779 df-trl 38833 df-tendo 39429 df-edring 39431 |
This theorem is referenced by: (None) |
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