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 38248 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | hdmapglem6.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | 5 | lmodring 19644 | . . . . 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 3950 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 1, 2, 5, 8, 9, 3, 11 | hgmapcl 39027 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵) |
13 | 1, 2, 5, 8, 9, 3, 12 | hgmapcl 39027 | . . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) |
14 | hdmapglem6.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) | |
15 | 14 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
16 | 1, 2, 5, 8, 9, 3, 15 | hgmapcl 39027 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵) |
17 | 1, 2, 3 | dvhlvec 38247 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
18 | 5 | lvecdrng 19879 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing) |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
20 | eldifsni 4724 | . . . . . . 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 39042 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 )) |
24 | 23 | necon3bid 3062 | . . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌) ≠ 0 ↔ 𝑌 ≠ 0 )) |
25 | 21, 24 | mpbird 259 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑌) ≠ 0 ) |
26 | hdmapglem6.n | . . . . . 6 ⊢ 𝑁 = (invr‘𝑅) | |
27 | 8, 22, 26 | drnginvrcl 19521 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
28 | 19, 16, 25, 27 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
29 | hdmapglem6.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
30 | 8, 29 | ringass 19316 | . . . 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 1368 | . . 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 19524 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
34 | 19, 16, 25, 33 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
35 | 34 | oveq2d 7174 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 )) |
36 | 8, 29, 32 | ringridm 19324 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
37 | 7, 13, 36 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
38 | 31, 35, 37 | 3eqtrrd 2863 | . 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 6676 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 )) |
41 | 1, 2, 5, 8, 29, 9, 3, 15, 12 | hgmapmul 39033 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
42 | 40, 41 | eqtr3d 2860 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
43 | hdmapglem6.cd | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) | |
44 | 43 | fveq2d 6676 | . . . . . 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 2823 | . . . . . . 7 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
49 | eqid 2823 | . . . . . . 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 39060 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
55 | 44, 54 | eqtr3d 2860 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷)) |
56 | 42, 55 | eqtr3d 2860 | . . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
57 | 39, 43 | eqtr4d 2861 | . . . . 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 39059 | . . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
59 | 56, 58 | eqtr4d 2861 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌))) |
60 | 59 | oveq1d 7173 | . 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
61 | 8, 29 | ringass 19316 | . . . 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 1368 | . . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
63 | 34 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 )) |
64 | 8, 29, 32 | ringridm 19324 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋) |
65 | 7, 11, 64 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋) |
66 | 62, 63, 65 | 3eqtrd 2862 | . 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋) |
67 | 38, 60, 66 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 〈cop 4575 I cid 5461 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 -gcsg 18107 1rcur 19253 Ringcrg 19299 invrcinvr 19423 DivRingcdr 19504 LModclmod 19636 LVecclvec 19876 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 DVecHcdvh 38216 ocHcoch 38485 HDMapchdma 38930 HGMapchg 39021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 df-hvmap 38895 df-hdmap1 38931 df-hdmap 38932 df-hgmap 39022 |
This theorem is referenced by: hgmapvvlem2 39062 |
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