hgmapvvlem1 - Mathbox for Norm Megill

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Theorem hgmapvvlem1 42511

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 41698	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmapglem6.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6	5	lmodring 20915	. . . . 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 3916	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	1, 2, 5, 8, 9, 3, 11	hgmapcl 42477	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵)
13	1, 2, 5, 8, 9, 3, 12	hgmapcl 42477	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵)
14		hdmapglem6.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
15	14	eldifad 3916	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	1, 2, 5, 8, 9, 3, 15	hgmapcl 42477	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵)
17	1, 2, 3	dvhlvec 41697	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18	5	lvecdrng 21152	. . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing)
20		eldifsni 4749	. . . . . . 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 42492	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 ))
24	23	necon3bid 3000	. . . . . 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 20782	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
28	19, 16, 25, 27	syl3anc 1389	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
29		hdmapglem6.t	. . . . 5 ⊢ × = (._r‘𝑅)
30	8, 29	ringass 20282	. . . 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 1390	. . 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 20786	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
34	19, 16, 25, 33	syl3anc 1389	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
35	34	oveq2d 7408	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 ))
36	8, 29, 32	ringridm 20299	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
37	7, 13, 36	syl2anc 593	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
38	31, 35, 37	3eqtrrd 2801	. 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 6867	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 ))
41	1, 2, 5, 8, 29, 9, 3, 15, 12	hgmapmul 42483	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
42	40, 41	eqtr3d 2798	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
43		hdmapglem6.cd	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
44	43	fveq2d 6867	. . . . . 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 2761	. . . . . . 7 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
49		eqid 2761	. . . . . . 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 42510	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
55	44, 54	eqtr3d 2798	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷))
56	42, 55	eqtr3d 2798	. . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
57	39, 43	eqtr4d 2799	. . . . 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 42509	. . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
59	56, 58	eqtr4d 2799	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌)))
60	59	oveq1d 7407	. 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
61	8, 29	ringass 20282	. . . 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 1390	. . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
63	34	oveq2d 7408	. . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 ))
64	8, 29, 32	ringridm 20299	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋)
65	7, 11, 64	syl2anc 593	. . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋)
66	62, 63, 65	3eqtrd 2800	. 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋)
67	38, 60, 66	3eqtrd 2800	1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3901 {csn 4581 ⟨cop 4587 I cid 5539 ↾ cres 5647 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 ._rcmulr 17270 Scalarcsca 17272 ·_𝑠 cvsca 17273 0_gc0g 17451 -_gcsg 18960 1_rcur 20210 Ringcrg 20262 inv_rcinvr 20415 DivRingcdr 20758 LModclmod 20907 LVecclvec 21149 HLchlt 39938 LHypclh 40572 LTrncltrn 40689 DVecHcdvh 41666 ocHcoch 41935 HDMapchdma 42380 HGMapchg 42471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 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 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-undef 8248 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mre 17597 df-mrc 17598 df-acs 17600 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-oppg 19369 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20542 df-rlreg 20723 df-domn 20724 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-lvec 21150 df-lsatoms 39564 df-lshyp 39565 df-lcv 39607 df-lfl 39646 df-lkr 39674 df-ldual 39712 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747 df-tgrp 41331 df-tendo 41343 df-edring 41345 df-dveca 41591 df-disoa 41617 df-dvech 41667 df-dib 41727 df-dic 41761 df-dih 41817 df-doch 41936 df-djh 41983 df-lcdual 42175 df-mapd 42213 df-hvmap 42345 df-hdmap1 42381 df-hdmap 42382 df-hgmap 42472

This theorem is referenced by: hgmapvvlem2 42512

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