hgmapvvlem1 - Mathbox for Norm Megill

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Theorem hgmapvvlem1 42428

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 41615	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmapglem6.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6	5	lmodring 20861	. . . . 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 3896	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	1, 2, 5, 8, 9, 3, 11	hgmapcl 42394	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵)
13	1, 2, 5, 8, 9, 3, 12	hgmapcl 42394	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵)
14		hdmapglem6.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
15	14	eldifad 3896	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	1, 2, 5, 8, 9, 3, 15	hgmapcl 42394	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵)
17	1, 2, 3	dvhlvec 41614	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18	5	lvecdrng 21098	. . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing)
20		eldifsni 4725	. . . . . . 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 42409	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 ))
24	23	necon3bid 2980	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌) ≠ 0 ↔ 𝑌 ≠ 0 ))
25	21, 24	mpbird 259	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑌) ≠ 0 )
26		hdmapglem6.n	. . . . . 6 ⊢ 𝑁 = (inv_r‘𝑅)
27	8, 22, 26	drnginvrcl 20728	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
28	19, 16, 25, 27	syl3anc 1380	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
29		hdmapglem6.t	. . . . 5 ⊢ × = (._r‘𝑅)
30	8, 29	ringass 20228	. . . 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 1381	. . 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 20732	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
34	19, 16, 25, 33	syl3anc 1380	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
35	34	oveq2d 7375	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 ))
36	8, 29, 32	ringridm 20245	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
37	7, 13, 36	syl2anc 591	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
38	31, 35, 37	3eqtrrd 2781	. 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 6834	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 ))
41	1, 2, 5, 8, 29, 9, 3, 15, 12	hgmapmul 42400	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
42	40, 41	eqtr3d 2778	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
43		hdmapglem6.cd	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
44	43	fveq2d 6834	. . . . . 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 2741	. . . . . . 7 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
49		eqid 2741	. . . . . . 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 42427	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
55	44, 54	eqtr3d 2778	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷))
56	42, 55	eqtr3d 2778	. . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
57	39, 43	eqtr4d 2779	. . . . 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 42426	. . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
59	56, 58	eqtr4d 2779	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌)))
60	59	oveq1d 7374	. 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
61	8, 29	ringass 20228	. . . 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 1381	. . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
63	34	oveq2d 7375	. . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 ))
64	8, 29, 32	ringridm 20245	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋)
65	7, 11, 64	syl2anc 591	. . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋)
66	62, 63, 65	3eqtrd 2780	. 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋)
67	38, 60, 66	3eqtrd 2780	1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∖ cdif 3881 {csn 4557 ⟨cop 4563 I cid 5514 ↾ cres 5622 ‘cfv 6488 (class class class)co 7359 Basecbs 17174 +_gcplusg 17215 ._rcmulr 17216 Scalarcsca 17218 ·_𝑠 cvsca 17219 0_gc0g 17397 -_gcsg 18906 1_rcur 20156 Ringcrg 20208 inv_rcinvr 20361 DivRingcdr 20704 LModclmod 20853 LVecclvec 21095 HLchlt 39855 LHypclh 40489 LTrncltrn 40606 DVecHcdvh 41583 ocHcoch 41852 HDMapchdma 42297 HGMapchg 42388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-riotaBAD 39458

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-0g 17399 df-mre 17543 df-mrc 17544 df-acs 17546 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18393 df-clat 18460 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-nzr 20488 df-rlreg 20669 df-domn 20670 df-drng 20706 df-lmod 20855 df-lss 20925 df-lsp 20965 df-lvec 21096 df-lsatoms 39481 df-lshyp 39482 df-lcv 39524 df-lfl 39563 df-lkr 39591 df-ldual 39629 df-oposet 39681 df-ol 39683 df-oml 39684 df-covers 39771 df-ats 39772 df-atl 39803 df-cvlat 39827 df-hlat 39856 df-llines 40003 df-lplanes 40004 df-lvols 40005 df-lines 40006 df-psubsp 40008 df-pmap 40009 df-padd 40301 df-lhyp 40493 df-laut 40494 df-ldil 40609 df-ltrn 40610 df-trl 40664 df-tgrp 41248 df-tendo 41260 df-edring 41262 df-dveca 41508 df-disoa 41534 df-dvech 41584 df-dib 41644 df-dic 41678 df-dih 41734 df-doch 41853 df-djh 41900 df-lcdual 42092 df-mapd 42130 df-hvmap 42262 df-hdmap1 42298 df-hdmap 42299 df-hgmap 42389

This theorem is referenced by: hgmapvvlem2 42429

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