hgmapvvlem1 - Mathbox for Norm Megill

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Theorem hgmapvvlem1 41884

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 41071	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmapglem6.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6	5	lmodring 20834	. . . . 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 3943	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	1, 2, 5, 8, 9, 3, 11	hgmapcl 41850	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵)
13	1, 2, 5, 8, 9, 3, 12	hgmapcl 41850	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵)
14		hdmapglem6.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
15	14	eldifad 3943	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	1, 2, 5, 8, 9, 3, 15	hgmapcl 41850	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵)
17	1, 2, 3	dvhlvec 41070	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18	5	lvecdrng 21072	. . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing)
20		eldifsni 4770	. . . . . . 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 41865	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 ))
24	23	necon3bid 2975	. . . . . 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 20721	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
28	19, 16, 25, 27	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
29		hdmapglem6.t	. . . . 5 ⊢ × = (._r‘𝑅)
30	8, 29	ringass 20218	. . . 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 1373	. . 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 20725	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
34	19, 16, 25, 33	syl3anc 1372	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
35	34	oveq2d 7429	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 ))
36	8, 29, 32	ringridm 20235	. . . 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 2774	. 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 6890	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 ))
41	1, 2, 5, 8, 29, 9, 3, 15, 12	hgmapmul 41856	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
42	40, 41	eqtr3d 2771	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
43		hdmapglem6.cd	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
44	43	fveq2d 6890	. . . . . 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 2734	. . . . . . 7 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
49		eqid 2734	. . . . . . 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 41883	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
55	44, 54	eqtr3d 2771	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷))
56	42, 55	eqtr3d 2771	. . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
57	39, 43	eqtr4d 2772	. . . . 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 41882	. . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
59	56, 58	eqtr4d 2772	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌)))
60	59	oveq1d 7428	. 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
61	8, 29	ringass 20218	. . . 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 1373	. . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
63	34	oveq2d 7429	. . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 ))
64	8, 29, 32	ringridm 20235	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋)
65	7, 11, 64	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋)
66	62, 63, 65	3eqtrd 2773	. 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋)
67	38, 60, 66	3eqtrd 2773	1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∖ cdif 3928 {csn 4606 ⟨cop 4612 I cid 5557 ↾ cres 5667 ‘cfv 6541 (class class class)co 7413 Basecbs 17229 +_gcplusg 17273 ._rcmulr 17274 Scalarcsca 17276 ·_𝑠 cvsca 17277 0_gc0g 17455 -_gcsg 18922 1_rcur 20146 Ringcrg 20198 inv_rcinvr 20355 DivRingcdr 20697 LModclmod 20826 LVecclvec 21069 HLchlt 39310 LHypclh 39945 LTrncltrn 40062 DVecHcdvh 41039 ocHcoch 41308 HDMapchdma 41753 HGMapchg 41844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-riotaBAD 38913

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-sca 17289 df-vsca 17290 df-0g 17457 df-mre 17600 df-mrc 17601 df-acs 17603 df-proset 18310 df-poset 18329 df-plt 18344 df-lub 18360 df-glb 18361 df-join 18362 df-meet 18363 df-p0 18439 df-p1 18440 df-lat 18446 df-clat 18513 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-subg 19110 df-cntz 19304 df-oppg 19333 df-lsm 19622 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-nzr 20481 df-rlreg 20662 df-domn 20663 df-drng 20699 df-lmod 20828 df-lss 20898 df-lsp 20938 df-lvec 21070 df-lsatoms 38936 df-lshyp 38937 df-lcv 38979 df-lfl 39018 df-lkr 39046 df-ldual 39084 df-oposet 39136 df-ol 39138 df-oml 39139 df-covers 39226 df-ats 39227 df-atl 39258 df-cvlat 39282 df-hlat 39311 df-llines 39459 df-lplanes 39460 df-lvols 39461 df-lines 39462 df-psubsp 39464 df-pmap 39465 df-padd 39757 df-lhyp 39949 df-laut 39950 df-ldil 40065 df-ltrn 40066 df-trl 40120 df-tgrp 40704 df-tendo 40716 df-edring 40718 df-dveca 40964 df-disoa 40990 df-dvech 41040 df-dib 41100 df-dic 41134 df-dih 41190 df-doch 41309 df-djh 41356 df-lcdual 41548 df-mapd 41586 df-hvmap 41718 df-hdmap1 41754 df-hdmap 41755 df-hgmap 41845

This theorem is referenced by: hgmapvvlem2 41885

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