tendoinvcl - Mathbox for Norm Megill

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Theorem tendoinvcl 38730

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 38609. (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 2738	. . . . . . 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 38708	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 = ((EDRing‘𝐾)‘𝑊))
6	1, 2	erngdv 38619	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
7	5, 6	eqeltrd 2833	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 ∈ DivRing)
8	7	3ad2ant1 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 ∈ DivRing)
9		simp2 1138	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ 𝐸)
10		tendoinv.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
11		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
12	1, 10, 3, 4, 11	dvhbase 38709	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐹) = 𝐸)
13	12	3ad2ant1 1134	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (Base‘𝐹) = 𝐸)
14	9, 13	eleqtrrd 2836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ (Base‘𝐹))
15		simp3 1139	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ 𝑂)
16	5	fveq2d 6672	. . . . . . 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 2738	. . . . . . . 8 ⊢ (0_g‘((EDRing‘𝐾)‘𝑊)) = (0_g‘((EDRing‘𝐾)‘𝑊))
21	17, 1, 18, 2, 19, 20	erng0g 38620	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘((EDRing‘𝐾)‘𝑊)) = 𝑂)
22	16, 21	eqtrd 2773	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = 𝑂)
23	22	3ad2ant1 1134	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (0_g‘𝐹) = 𝑂)
24	15, 23	neeqtrrd 3008	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ (0_g‘𝐹))
25		eqid 2738	. . . . 5 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
26		tendoinv.n	. . . . 5 ⊢ 𝑁 = (inv_r‘𝐹)
27	11, 25, 26	drnginvrcl 19631	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ∈ (Base‘𝐹))
28	8, 14, 24, 27	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ (Base‘𝐹))
29	28, 13	eleqtrd 2835	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ 𝐸)
30	11, 25, 26	drnginvrn0 19632	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
31	8, 14, 24, 30	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
32	31, 23	neeqtrd 3003	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ 𝑂)
33	29, 32	jca 515	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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ↦ cmpt 5107 I cid 5424 ↾ cres 5521 ‘cfv 6333 Basecbs 16579 Scalarcsca 16664 0_gc0g 16809 inv_rcinvr 19536 DivRingcdr 19614 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 TEndoctendo 38378 EDRingcedring 38379 DVecHcdvh 38704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-minusg 18216 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tendo 38381 df-edring 38383 df-dvech 38705

This theorem is referenced by: tendolinv 38731 tendorinv 38732 dih1dimatlem0 38954 dih1dimatlem 38955

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