erngdvlem1 - Mathbox for Norm Megill

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Theorem erngdvlem1 41456

Description: Lemma for eringring 41460. (Contributed by NM, 4-Aug-2013.)

**Hypotheses**
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)))

**Assertion**
Ref	Expression
erngdvlem1	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)

Distinct variable groups: 𝐵,𝑓 𝑎,𝑏,𝐸 𝑓,𝑎,𝐾,𝑏 𝑓,𝐻 𝑇,𝑎,𝑏,𝑓 𝑊,𝑎,𝑏,𝑓 Allowed substitution hints: 𝐵(𝑎,𝑏) 𝐷(𝑓,𝑎,𝑏) 𝑃(𝑓,𝑎,𝑏) 𝐸(𝑓) 𝐻(𝑎,𝑏) 𝐼(𝑓,𝑎,𝑏) 0 (𝑓,𝑎,𝑏)

Proof of Theorem erngdvlem1 Dummy variables 𝑡 𝑠 𝑢 are mutually distinct and distinct from all other variables.

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 2737	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	1, 2, 3, 4, 5	erngbase 41269	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
7	6	eqcomd 2743	. 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 2737	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
10	1, 2, 3, 4, 9	erngfplus 41270	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))))
11	8, 10	eqtr4id 2791	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+_g‘𝐷))
12	1, 2, 3, 8	tendoplcl 41249	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸)
13	1, 2, 3, 8	tendoplass 41251	. 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 41258	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
17	14, 1, 2, 3, 15, 8	tendo0pl 41259	. 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 41264	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐼‘𝑠) ∈ 𝐸)
20	1, 2, 3, 18, 14, 8, 15	tendoipl 41265	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ((𝐼‘𝑠)𝑃𝑠) = 0 )
21	7, 11, 12, 13, 16, 17, 19, 20	isgrpd 18931	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5167 I cid 5522 ^◡ccnv 5627 ↾ cres 5630 ∘ ccom 5632 ‘cfv 6496 ∈ cmpo 7366 Basecbs 17176 +_gcplusg 17217 Grpcgrp 18906 HLchlt 39818 LHypclh 40452 LTrncltrn 40569 TEndoctendo 41220 EDRingcedring 41221

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-riotaBAD 39421

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-n0 12435 df-z 12522 df-uz 12786 df-fz 13459 df-struct 17114 df-slot 17149 df-ndx 17161 df-base 17177 df-plusg 17230 df-mulr 17231 df-0g 17401 df-proset 18257 df-poset 18276 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18395 df-clat 18462 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18909 df-oposet 39644 df-ol 39646 df-oml 39647 df-covers 39734 df-ats 39735 df-atl 39766 df-cvlat 39790 df-hlat 39819 df-llines 39966 df-lplanes 39967 df-lvols 39968 df-lines 39969 df-psubsp 39971 df-pmap 39972 df-padd 40264 df-lhyp 40456 df-laut 40457 df-ldil 40572 df-ltrn 40573 df-trl 40627 df-tendo 41223 df-edring 41225

This theorem is referenced by: erngdvlem2N 41457 erngdvlem3 41458 erngdvlem4 41459

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