tendoinvcl - Mathbox for Norm Megill

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Theorem tendoinvcl 40805

Description: Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 40684. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)

**Hypotheses**
Ref	Expression
tendoinv.b	⊢ 𝐵 = (Base‘𝐾)
tendoinv.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoinv.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoinv.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
tendoinv.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
tendoinv.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
tendoinv.f	⊢ 𝐹 = (Scalar‘𝑈)
tendoinv.n	⊢ 𝑁 = (inv_r‘𝐹)

**Assertion**
Ref	Expression
tendoinvcl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂))

Distinct variable groups: 𝐵,ℎ ℎ,𝐻 ℎ,𝐾 𝑇,ℎ ℎ,𝑊 Allowed substitution hints: 𝑆(ℎ) 𝑈(ℎ) 𝐸(ℎ) 𝐹(ℎ) 𝑁(ℎ) 𝑂(ℎ)

**Proof of Theorem tendoinvcl**
Step	Hyp	Ref	Expression
1		tendoinv.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2726	. . . . . . 7 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
3		tendoinv.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		tendoinv.f	. . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑈)
5	1, 2, 3, 4	dvhsca 40783	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 = ((EDRing‘𝐾)‘𝑊))
6	1, 2	erngdv 40694	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
7	5, 6	eqeltrd 2826	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 ∈ DivRing)
8	7	3ad2ant1 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 ∈ DivRing)
9		simp2 1134	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ 𝐸)
10		tendoinv.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
11		eqid 2726	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
12	1, 10, 3, 4, 11	dvhbase 40784	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐹) = 𝐸)
13	12	3ad2ant1 1130	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (Base‘𝐹) = 𝐸)
14	9, 13	eleqtrrd 2829	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ (Base‘𝐹))
15		simp3 1135	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ 𝑂)
16	5	fveq2d 6907	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = (0_g‘((EDRing‘𝐾)‘𝑊)))
17		tendoinv.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
18		tendoinv.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
19		tendoinv.o	. . . . . . . 8 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
20		eqid 2726	. . . . . . . 8 ⊢ (0_g‘((EDRing‘𝐾)‘𝑊)) = (0_g‘((EDRing‘𝐾)‘𝑊))
21	17, 1, 18, 2, 19, 20	erng0g 40695	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘((EDRing‘𝐾)‘𝑊)) = 𝑂)
22	16, 21	eqtrd 2766	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = 𝑂)
23	22	3ad2ant1 1130	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (0_g‘𝐹) = 𝑂)
24	15, 23	neeqtrrd 3005	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ (0_g‘𝐹))
25		eqid 2726	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
26		tendoinv.n	. . . . 5 ⊢ 𝑁 = (inv_r‘𝐹)
27	11, 25, 26	drnginvrcl 20733	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ∈ (Base‘𝐹))
28	8, 14, 24, 27	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ (Base‘𝐹))
29	28, 13	eleqtrd 2828	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ 𝐸)
30	11, 25, 26	drnginvrn0 20734	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
31	8, 14, 24, 30	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
32	31, 23	neeqtrd 3000	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ 𝑂)
33	29, 32	jca 510	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ↦ cmpt 5238 I cid 5581 ↾ cres 5686 ‘cfv 6556 Basecbs 17215 Scalarcsca 17271 0_gc0g 17456 inv_rcinvr 20371 DivRingcdr 20709 HLchlt 39050 LHypclh 39685 LTrncltrn 39802 TEndoctendo 40453 EDRingcedring 40454 DVecHcdvh 40779

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 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-riotaBAD 38653

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 4916 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-undef 8290 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-0g 17458 df-proset 18322 df-poset 18340 df-plt 18357 df-lub 18373 df-glb 18374 df-join 18375 df-meet 18376 df-p0 18452 df-p1 18453 df-lat 18459 df-clat 18526 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-dvr 20385 df-drng 20711 df-oposet 38876 df-ol 38878 df-oml 38879 df-covers 38966 df-ats 38967 df-atl 38998 df-cvlat 39022 df-hlat 39051 df-llines 39199 df-lplanes 39200 df-lvols 39201 df-lines 39202 df-psubsp 39204 df-pmap 39205 df-padd 39497 df-lhyp 39689 df-laut 39690 df-ldil 39805 df-ltrn 39806 df-trl 39860 df-tendo 40456 df-edring 40458 df-dvech 40780

This theorem is referenced by: tendolinv 40806 tendorinv 40807 dih1dimatlem0 41029 dih1dimatlem 41030

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