hgmapvvlem1 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapvvlem1		Structured version Visualization version GIF version

Theorem hgmapvvlem1 42293

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 41480	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		hdmapglem6.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6	5	lmodring 20831	. . . . 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 3915	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	1, 2, 5, 8, 9, 3, 11	hgmapcl 42259	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵)
13	1, 2, 5, 8, 9, 3, 12	hgmapcl 42259	. . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵)
14		hdmapglem6.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
15	14	eldifad 3915	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	1, 2, 5, 8, 9, 3, 15	hgmapcl 42259	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵)
17	1, 2, 3	dvhlvec 41479	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
18	5	lvecdrng 21069	. . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing)
20		eldifsni 4748	. . . . . . 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 42274	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 ))
24	23	necon3bid 2977	. . . . . 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 20698	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
28	19, 16, 25, 27	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)
29		hdmapglem6.t	. . . . 5 ⊢ × = (._r‘𝑅)
30	8, 29	ringass 20200	. . . 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 1375	. . 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 20702	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
34	19, 16, 25, 33	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 )
35	34	oveq2d 7384	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 ))
36	8, 29, 32	ringridm 20217	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
37	7, 13, 36	syl2anc 585	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋)))
38	31, 35, 37	3eqtrrd 2777	. 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 6846	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 ))
41	1, 2, 5, 8, 29, 9, 3, 15, 12	hgmapmul 42265	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
42	40, 41	eqtr3d 2774	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)))
43		hdmapglem6.cd	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )
44	43	fveq2d 6846	. . . . . 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 2737	. . . . . . 7 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
49		eqid 2737	. . . . . . 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 42292	. . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
55	44, 54	eqtr3d 2774	. . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷))
56	42, 55	eqtr3d 2774	. . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
57	39, 43	eqtr4d 2775	. . . . 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 42291	. . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷))
59	56, 58	eqtr4d 2775	. . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌)))
60	59	oveq1d 7383	. 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))))
61	8, 29	ringass 20200	. . . 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 1375	. . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))))
63	34	oveq2d 7384	. . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 ))
64	8, 29, 32	ringridm 20217	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋)
65	7, 11, 64	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋)
66	62, 63, 65	3eqtrd 2776	. 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋)
67	38, 60, 66	3eqtrd 2776	1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 {csn 4582 ⟨cop 4588 I cid 5526 ↾ cres 5634 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 Scalarcsca 17192 ·_𝑠 cvsca 17193 0_gc0g 17371 -_gcsg 18877 1_rcur 20128 Ringcrg 20180 inv_rcinvr 20335 DivRingcdr 20674 LModclmod 20823 LVecclvec 21066 HLchlt 39720 LHypclh 40354 LTrncltrn 40471 DVecHcdvh 41448 ocHcoch 41717 HDMapchdma 42162 HGMapchg 42253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-nzr 20458 df-rlreg 20639 df-domn 20640 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lshyp 39347 df-lcv 39389 df-lfl 39428 df-lkr 39456 df-ldual 39494 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tgrp 41113 df-tendo 41125 df-edring 41127 df-dveca 41373 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718 df-djh 41765 df-lcdual 41957 df-mapd 41995 df-hvmap 42127 df-hdmap1 42163 df-hdmap 42164 df-hgmap 42254

This theorem is referenced by: hgmapvvlem2 42294

W3C validator