hgmapvvlem1 - Mathbox for Norm Megill

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Theorem hgmapvvlem1 42179

Description: Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our 𝐸, 𝐶, 𝐷, 𝑌, 𝑋 correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapglem6.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapglem6.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapglem6.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapglem6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapglem6.v	⊢ 𝑉 = (Base‘𝑈)
hdmapglem6.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapglem6.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapglem6.b	⊢ 𝐵 = (Base‘𝑅)
hdmapglem6.t	⊢ × = (._r‘𝑅)
hdmapglem6.z	⊢ 0 = (0_g‘𝑅)
hdmapglem6.i	⊢ 1 = (1_r‘𝑅)
hdmapglem6.n	⊢ 𝑁 = (inv_r‘𝑅)
hdmapglem6.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapglem6.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapglem6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapglem6.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))
hdmapglem6.c	⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
hdmapglem6.d	⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
hdmapglem6.cd	⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
hdmapglem6.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
hdmapglem6.yx	⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 )

**Assertion**
Ref	Expression
hgmapvvlem1	⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

**Proof of Theorem hgmapvvlem1**
Step	Hyp	Ref	Expression
1		hdmapglem6.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapglem6.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmapglem6.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 41366	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmapglem6.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6	5	lmodring 20819	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
7	4, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
8		hdmapglem6.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
9		hdmapglem6.g	. . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
10		hdmapglem6.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))
11	10	eldifad 3913	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	1, 2, 5, 8, 9, 3, 11	hgmapcl 42145	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵)
13	1, 2, 5, 8, 9, 3, 12	hgmapcl 42145	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵)
14		hdmapglem6.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
15	14	eldifad 3913	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	1, 2, 5, 8, 9, 3, 15	hgmapcl 42145	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵)
17	1, 2, 3	dvhlvec 41365	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18	5	lvecdrng 21057	. . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing)
20		eldifsni 4746	. . . . . . 7 ⊢ (𝑌 ∈ (𝐵 ∖ { 0 }) → 𝑌 ≠ 0 )
21	14, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 )
22		hdmapglem6.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑅)
23	1, 2, 5, 8, 22, 9, 3, 15	hgmapeq0 42160	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 ))
24	23	necon3bid 2976	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌) ≠ 0 ↔ 𝑌 ≠ 0 ))
25	21, 24	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑌) ≠ 0 )
26		hdmapglem6.n	. . . . . 6 ⊢ 𝑁 = (inv_r‘𝑅)
27	8, 22, 26	drnginvrcl 20686	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
28	19, 16, 25, 27	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
29		hdmapglem6.t	. . . . 5 ⊢ × = (._r‘𝑅)
30	8, 29	ringass 20188	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝐺‘(𝐺‘𝑋)) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)) → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
31	7, 13, 16, 28, 30	syl13anc 1374	. . 3 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
32		hdmapglem6.i	. . . . . 6 ⊢ 1 = (1_r‘𝑅)
33	8, 22, 29, 32, 26	drnginvrr 20690	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
34	19, 16, 25, 33	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
35	34	oveq2d 7374	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 ))
36	8, 29, 32	ringridm 20205	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
37	7, 13, 36	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
38	31, 35, 37	3eqtrrd 2776	. 2 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
39		hdmapglem6.yx	. . . . . . 7 ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 )
40	39	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 ))
41	1, 2, 5, 8, 29, 9, 3, 15, 12	hgmapmul 42151	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
42	40, 41	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
43		hdmapglem6.cd	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
44	43	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = (𝐺‘ 1 ))
45		hdmapglem6.e	. . . . . . 7 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
46		hdmapglem6.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
47		hdmapglem6.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
48		eqid 2736	. . . . . . 7 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
49		eqid 2736	. . . . . . 7 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
50		hdmapglem6.q	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
51		hdmapglem6.s	. . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
52		hdmapglem6.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
53		hdmapglem6.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
54	1, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11	hdmapglem5 42178	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
55	44, 54	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷))
56	42, 55	eqtr3d 2773	. . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
57	39, 43	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = ((𝑆‘𝐷)‘𝐶))
58	1, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11, 57	hdmapinvlem4 42177	. . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
59	56, 58	eqtr4d 2774	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌)))
60	59	oveq1d 7373	. 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
61	8, 29	ringass 20188	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)) → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
62	7, 11, 16, 28, 61	syl13anc 1374	. . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
63	34	oveq2d 7374	. . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 ))
64	8, 29, 32	ringridm 20205	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋)
65	7, 11, 64	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋)
66	62, 63, 65	3eqtrd 2775	. 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋)
67	38, 60, 66	3eqtrd 2775	1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 {csn 4580 ⟨cop 4586 I cid 5518 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 ._rcmulr 17178 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 -_gcsg 18865 1_rcur 20116 Ringcrg 20168 inv_rcinvr 20323 DivRingcdr 20662 LModclmod 20811 LVecclvec 21054 HLchlt 39606 LHypclh 40240 LTrncltrn 40357 DVecHcdvh 41334 ocHcoch 41603 HDMapchdma 42048 HGMapchg 42139

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39209

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-ot 4589 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-rlreg 20627 df-domn 20628 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39232 df-lshyp 39233 df-lcv 39275 df-lfl 39314 df-lkr 39342 df-ldual 39380 df-oposet 39432 df-ol 39434 df-oml 39435 df-covers 39522 df-ats 39523 df-atl 39554 df-cvlat 39578 df-hlat 39607 df-llines 39754 df-lplanes 39755 df-lvols 39756 df-lines 39757 df-psubsp 39759 df-pmap 39760 df-padd 40052 df-lhyp 40244 df-laut 40245 df-ldil 40360 df-ltrn 40361 df-trl 40415 df-tgrp 40999 df-tendo 41011 df-edring 41013 df-dveca 41259 df-disoa 41285 df-dvech 41335 df-dib 41395 df-dic 41429 df-dih 41485 df-doch 41604 df-djh 41651 df-lcdual 41843 df-mapd 41881 df-hvmap 42013 df-hdmap1 42049 df-hdmap 42050 df-hgmap 42140

This theorem is referenced by: hgmapvvlem2 42180

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