tendolinv - Mathbox for Norm Megill

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Theorem tendolinv 41729

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 1149	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		tendoinv.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2762	. . . . . 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 41706	. . . . 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 41617	. . . . 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 2862	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝐹 ∈ DivRing)
11		simp2 1150	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ 𝐸)
12		tendoinv.e	. . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
13		eqid 2762	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
14	2, 12, 4, 5, 13	dvhbase 41707	. . . . 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 2865	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ∈ (Base‘𝐹))
17		simp3 1151	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ 𝑂)
18	6	fveq2d 6871	. . . . . 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 2762	. . . . . . 7 ⊢ (0_g‘((EDRing‘𝐾)‘𝑊)) = (0_g‘((EDRing‘𝐾)‘𝑊))
23	19, 2, 20, 3, 21, 22	erng0g 41618	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘((EDRing‘𝐾)‘𝑊)) = 𝑂)
24	18, 23	eqtrd 2797	. . . . 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 3031	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → 𝑆 ≠ (0_g‘𝐹))
27		eqid 2762	. . . 4 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
28		eqid 2762	. . . 4 ⊢ (._r‘𝐹) = (._r‘𝐹)
29		eqid 2762	. . . 4 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
30		tendoinv.n	. . . 4 ⊢ 𝑁 = (inv_r‘𝐹)
31	13, 27, 28, 29, 30	drnginvrl 20806	. . 3 ⊢ ((𝐹 ∈ DivRing ∧ 𝑆 ∈ (Base‘𝐹) ∧ 𝑆 ≠ (0_g‘𝐹)) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = (1_r‘𝐹))
32	10, 16, 26, 31	syl3anc 1390	. 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 41728	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂))
34	33	simpld 498	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑁‘𝑆) ∈ 𝐸)
35	2, 20, 12, 4, 5, 28	dvhmulr 41710	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑁‘𝑆) ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = ((𝑁‘𝑆) ∘ 𝑆))
36	1, 34, 11, 35	syl12anc 847	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆)(._r‘𝐹)𝑆) = ((𝑁‘𝑆) ∘ 𝑆))
37	6	fveq2d 6871	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘𝐹) = (1_r‘((EDRing‘𝐾)‘𝑊)))
38		eqid 2762	. . . . 5 ⊢ (1_r‘((EDRing‘𝐾)‘𝑊)) = (1_r‘((EDRing‘𝐾)‘𝑊))
39	2, 20, 3, 38	erng1r 41619	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
40	37, 39	eqtrd 2797	. . 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 2805	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 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ↦ cmpt 5181 I cid 5541 ↾ cres 5649 ∘ ccom 5651 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 ._rcmulr 17287 Scalarcsca 17289 0_gc0g 17468 1_rcur 20231 inv_rcinvr 20436 DivRingcdr 20779 HLchlt 39974 LHypclh 40608 LTrncltrn 40725 TEndoctendo 41376 EDRingcedring 41377 DVecHcdvh 41702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-riotaBAD 39577

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-0g 17470 df-proset 18326 df-poset 18345 df-plt 18360 df-lub 18376 df-glb 18377 df-join 18378 df-meet 18379 df-p0 18455 df-p1 18456 df-lat 18464 df-clat 18531 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-drng 20781 df-oposet 39800 df-ol 39802 df-oml 39803 df-covers 39890 df-ats 39891 df-atl 39922 df-cvlat 39946 df-hlat 39975 df-llines 40122 df-lplanes 40123 df-lvols 40124 df-lines 40125 df-psubsp 40127 df-pmap 40128 df-padd 40420 df-lhyp 40612 df-laut 40613 df-ldil 40728 df-ltrn 40729 df-trl 40783 df-tendo 41379 df-edring 41381 df-dvech 41703

This theorem is referenced by: dih1dimatlem0 41952

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