tendoinvcl - Mathbox for Norm Megill

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Theorem tendoinvcl 39118

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 38997. (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 39096	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 = ((EDRing‘𝐾)‘𝑊))
6	1, 2	erngdv 39007	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
7	5, 6	eqeltrd 2839	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 ∈ DivRing)
8	7	3ad2ant1 1132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 ∈ DivRing)
9		simp2 1136	. . . . 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 39097	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐹) = 𝐸)
13	12	3ad2ant1 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (Base‘𝐹) = 𝐸)
14	9, 13	eleqtrrd 2842	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ (Base‘𝐹))
15		simp3 1137	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ 𝑂)
16	5	fveq2d 6778	. . . . . . 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 39008	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘((EDRing‘𝐾)‘𝑊)) = 𝑂)
22	16, 21	eqtrd 2778	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = 𝑂)
23	22	3ad2ant1 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (0_g‘𝐹) = 𝑂)
24	15, 23	neeqtrrd 3018	. . . 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 20008	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ∈ (Base‘𝐹))
28	8, 14, 24, 27	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ (Base‘𝐹))
29	28, 13	eleqtrd 2841	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ 𝐸)
30	11, 25, 26	drnginvrn0 20009	. . . 4 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
31	8, 14, 24, 30	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ (0_g‘𝐹))
32	31, 23	neeqtrd 3013	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ≠ 𝑂)
33	29, 32	jca 512	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ↦ cmpt 5157 I cid 5488 ↾ cres 5591 ‘cfv 6433 Basecbs 16912 Scalarcsca 16965 0_gc0g 17150 inv_rcinvr 19913 DivRingcdr 19991 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 TEndoctendo 38766 EDRingcedring 38767 DVecHcdvh 39092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-edring 38771 df-dvech 39093

This theorem is referenced by: tendolinv 39119 tendorinv 39120 dih1dimatlem0 39342 dih1dimatlem 39343

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