Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapvvlem1 | Structured version Visualization version GIF version |
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.) |
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 = (0g‘𝑅) |
hdmapglem6.i | ⊢ 1 = (1r‘𝑅) |
hdmapglem6.n | ⊢ 𝑁 = (invr‘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 ) |
Ref | Expression |
---|---|
hgmapvvlem1 | ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
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 36399 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | hdmapglem6.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | 5 | lmodring 18871 | . . . . 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 3586 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 1, 2, 5, 8, 9, 3, 11 | hgmapcl 37181 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵) |
13 | 1, 2, 5, 8, 9, 3, 12 | hgmapcl 37181 | . . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) |
14 | hdmapglem6.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) | |
15 | 14 | eldifad 3586 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
16 | 1, 2, 5, 8, 9, 3, 15 | hgmapcl 37181 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵) |
17 | 1, 2, 3 | dvhlvec 36398 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
18 | 5 | lvecdrng 19105 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing) |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
20 | eldifsni 4320 | . . . . . . 7 ⊢ (𝑌 ∈ (𝐵 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
21 | 14, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
22 | hdmapglem6.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
23 | 1, 2, 5, 8, 22, 9, 3, 15 | hgmapeq0 37196 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 )) |
24 | 23 | necon3bid 2838 | . . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌) ≠ 0 ↔ 𝑌 ≠ 0 )) |
25 | 21, 24 | mpbird 247 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑌) ≠ 0 ) |
26 | hdmapglem6.n | . . . . . 6 ⊢ 𝑁 = (invr‘𝑅) | |
27 | 8, 22, 26 | drnginvrcl 18764 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
28 | 19, 16, 25, 27 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
29 | hdmapglem6.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
30 | 8, 29 | ringass 18564 | . . . 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 1328 | . . 3 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
32 | hdmapglem6.i | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
33 | 8, 22, 29, 32, 26 | drnginvrr 18767 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
34 | 19, 16, 25, 33 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
35 | 34 | oveq2d 6666 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 )) |
36 | 8, 29, 32 | ringridm 18572 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
37 | 7, 13, 36 | syl2anc 693 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
38 | 31, 35, 37 | 3eqtrrd 2661 | . 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 6195 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 )) |
41 | 1, 2, 5, 8, 29, 9, 3, 15, 12 | hgmapmul 37187 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
42 | 40, 41 | eqtr3d 2658 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
43 | hdmapglem6.cd | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) | |
44 | 43 | fveq2d 6195 | . . . . . 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 2622 | . . . . . . 7 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
49 | eqid 2622 | . . . . . . 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 37214 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
55 | 44, 54 | eqtr3d 2658 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷)) |
56 | 42, 55 | eqtr3d 2658 | . . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
57 | 39, 43 | eqtr4d 2659 | . . . . 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 37213 | . . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
59 | 56, 58 | eqtr4d 2659 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌))) |
60 | 59 | oveq1d 6665 | . 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
61 | 8, 29 | ringass 18564 | . . . 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 1328 | . . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
63 | 34 | oveq2d 6666 | . . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 )) |
64 | 8, 29, 32 | ringridm 18572 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋) |
65 | 7, 11, 64 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋) |
66 | 62, 63, 65 | 3eqtrd 2660 | . 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋) |
67 | 38, 60, 66 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 〈cop 4183 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 -gcsg 17424 1rcur 18501 Ringcrg 18547 invrcinvr 18671 DivRingcdr 18747 LModclmod 18863 LVecclvec 19102 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 DVecHcdvh 36367 ocHcoch 36636 HDMapchdma 37082 HGMapchg 37175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hvmap 37046 df-hdmap1 37083 df-hdmap 37084 df-hgmap 37176 |
This theorem is referenced by: hgmapvvlem2 37216 |
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