tendolinv - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > tendolinv		Structured version Visualization version GIF version

Theorem tendolinv 41304

Description: Left multiplicative inverse for endomorphism. (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
tendolinv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∘ 𝑆) = ( I ↾ 𝑇))

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 tendolinv**
Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		tendoinv.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2734	. . . . . 6 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
4		tendoinv.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		tendoinv.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑈)
6	2, 3, 4, 5	dvhsca 41281	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐹 = ((EDRing‘𝐾)‘𝑊))
7	1, 6	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 = ((EDRing‘𝐾)‘𝑊))
8	2, 3	erngdv 41192	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
9	1, 8	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
10	7, 9	eqeltrd 2834	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 ∈ DivRing)
11		simp2 1137	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ 𝐸)
12		tendoinv.e	. . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
13		eqid 2734	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
14	2, 12, 4, 5, 13	dvhbase 41282	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐹) = 𝐸)
15	1, 14	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (Base‘𝐹) = 𝐸)
16	11, 15	eleqtrrd 2837	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ (Base‘𝐹))
17		simp3 1138	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ 𝑂)
18	6	fveq2d 6836	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = (0_g‘((EDRing‘𝐾)‘𝑊)))
19		tendoinv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
20		tendoinv.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
21		tendoinv.o	. . . . . . 7 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
22		eqid 2734	. . . . . . 7 ⊢ (0_g‘((EDRing‘𝐾)‘𝑊)) = (0_g‘((EDRing‘𝐾)‘𝑊))
23	19, 2, 20, 3, 21, 22	erng0g 41193	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘((EDRing‘𝐾)‘𝑊)) = 𝑂)
24	18, 23	eqtrd 2769	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐹) = 𝑂)
25	1, 24	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (0_g‘𝐹) = 𝑂)
26	17, 25	neeqtrrd 3004	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ (0_g‘𝐹))
27		eqid 2734	. . . 4 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
28		eqid 2734	. . . 4 ⊢ (._r‘𝐹) = (._r‘𝐹)
29		eqid 2734	. . . 4 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
30		tendoinv.n	. . . 4 ⊢ 𝑁 = (inv_r‘𝐹)
31	13, 27, 28, 29, 30	drnginvrl 20687	. . 3 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = (1_r‘𝐹))
32	10, 16, 26, 31	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = (1_r‘𝐹))
33	19, 2, 20, 12, 21, 4, 5, 30	tendoinvcl 41303	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂))
34	33	simpld 494	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ 𝐸)
35	2, 20, 12, 4, 5, 28	dvhmulr 41285	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑁‘𝑆) ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = ((𝑁‘𝑆) ∘ 𝑆))
36	1, 34, 11, 35	syl12anc 836	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = ((𝑁‘𝑆) ∘ 𝑆))
37	6	fveq2d 6836	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘𝐹) = (1_r‘((EDRing‘𝐾)‘𝑊)))
38		eqid 2734	. . . . 5 ⊢ (1_r‘((EDRing‘𝐾)‘𝑊)) = (1_r‘((EDRing‘𝐾)‘𝑊))
39	2, 20, 3, 38	erng1r 41194	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
40	37, 39	eqtrd 2769	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘𝐹) = ( I ↾ 𝑇))
41	1, 40	syl 17	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (1_r‘𝐹) = ( I ↾ 𝑇))
42	32, 36, 41	3eqtr3d 2777	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∘ 𝑆) = ( I ↾ 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ↦ cmpt 5177 I cid 5516 ↾ cres 5624 ∘ ccom 5626 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 ._rcmulr 17176 Scalarcsca 17178 0_gc0g 17357 1_rcur 20114 inv_rcinvr 20321 DivRingcdr 20660 HLchlt 39549 LHypclh 40183 LTrncltrn 40300 TEndoctendo 40951 EDRingcedring 40952 DVecHcdvh 41277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-drng 20662 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tendo 40954 df-edring 40956 df-dvech 41278

This theorem is referenced by: dih1dimatlem0 41527

W3C validator