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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem7b | Structured version Visualization version GIF version |
Description: Lemma for hdmapg 37998. (Contributed by NM, 14-Jun-2015.) |
Ref | Expression |
---|---|
hdmapglem7.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapglem7.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapglem7.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
hdmapglem7.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapglem7.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapglem7.p | ⊢ + = (+g‘𝑈) |
hdmapglem7.q | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmapglem7.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapglem7.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapglem7.a | ⊢ ⊕ = (LSSum‘𝑈) |
hdmapglem7.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmapglem7.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapglem7.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmapglem7.t | ⊢ × = (.r‘𝑅) |
hdmapglem7.z | ⊢ 0 = (0g‘𝑅) |
hdmapglem7.c | ⊢ ✚ = (+g‘𝑅) |
hdmapglem7.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapglem7.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hdmapglem7b.u | ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸})) |
hdmapglem7b.v | ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸})) |
hdmapglem7b.k | ⊢ (𝜑 → 𝑚 ∈ 𝐵) |
hdmapglem7b.l | ⊢ (𝜑 → 𝑛 ∈ 𝐵) |
Ref | Expression |
---|---|
hdmapglem7b | ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglem7.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapglem7.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapglem7.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmapglem7.p | . . 3 ⊢ + = (+g‘𝑈) | |
5 | hdmapglem7.q | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
6 | hdmapglem7.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
7 | hdmapglem7.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
8 | hdmapglem7.c | . . 3 ⊢ ✚ = (+g‘𝑅) | |
9 | hdmapglem7.t | . . 3 ⊢ × = (.r‘𝑅) | |
10 | hdmapglem7.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hdmapglem7.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
12 | hdmapglem7.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | 1, 2, 12 | dvhlmod 37178 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
14 | hdmapglem7b.l | . . . . 5 ⊢ (𝜑 → 𝑛 ∈ 𝐵) | |
15 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2825 | . . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
17 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
18 | hdmapglem7.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | 1, 15, 16, 2, 3, 17, 18, 12 | dvheveccl 37180 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
20 | 19 | eldifad 3810 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
21 | 3, 6, 5, 7 | lmodvscl 19236 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝑛 · 𝐸) ∈ 𝑉) |
22 | 13, 14, 20, 21 | syl3anc 1494 | . . . 4 ⊢ (𝜑 → (𝑛 · 𝐸) ∈ 𝑉) |
23 | 20 | snssd 4558 | . . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
24 | hdmapglem7.o | . . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
25 | 1, 2, 3, 24 | dochssv 37423 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉) |
26 | 12, 23, 25 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉) |
27 | hdmapglem7b.v | . . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸})) | |
28 | 26, 27 | sseldd 3828 | . . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝑉) |
29 | 3, 4 | lmodvacl 19233 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑛 · 𝐸) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉) |
30 | 13, 22, 28, 29 | syl3anc 1494 | . . 3 ⊢ (𝜑 → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉) |
31 | hdmapglem7b.u | . . . 4 ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸})) | |
32 | 26, 31 | sseldd 3828 | . . 3 ⊢ (𝜑 → 𝑥 ∈ 𝑉) |
33 | hdmapglem7b.k | . . 3 ⊢ (𝜑 → 𝑚 ∈ 𝐵) | |
34 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 20, 32, 33 | hdmapgln2 37980 | . 2 ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 28, 20, 14 | hdmapln1 37974 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦))) |
36 | eqid 2825 | . . . . . . . . 9 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊) | |
37 | eqid 2825 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
38 | 1, 18, 36, 10, 12, 2, 6, 37 | hdmapevec2 37904 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1r‘𝑅)) |
39 | 38 | oveq2d 6921 | . . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1r‘𝑅))) |
40 | 6 | lmodring 19227 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring) |
41 | 13, 40 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) |
42 | 7, 9, 37 | ringridm 18926 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × (1r‘𝑅)) = 𝑛) |
43 | 41, 14, 42 | syl2anc 579 | . . . . . . 7 ⊢ (𝜑 → (𝑛 × (1r‘𝑅)) = 𝑛) |
44 | 39, 43 | eqtrd 2861 | . . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = 𝑛) |
45 | hdmapglem7.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
46 | 1, 18, 24, 2, 3, 6, 7, 9, 45, 10, 12, 27 | hdmapinvlem1 37986 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 ) |
47 | 44, 46 | oveq12d 6923 | . . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 )) |
48 | ringgrp 18906 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
49 | 41, 48 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
50 | 7, 8, 45 | grprid 17807 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵) → (𝑛 ✚ 0 ) = 𝑛) |
51 | 49, 14, 50 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑛 ✚ 0 ) = 𝑛) |
52 | 35, 47, 51 | 3eqtrd 2865 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = 𝑛) |
53 | 52 | oveq1d 6920 | . . 3 ⊢ (𝜑 → (((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) = (𝑛 × (𝐺‘𝑚))) |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 28, 32, 14 | hdmapln1 37974 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦))) |
55 | 1, 18, 24, 2, 3, 6, 7, 9, 45, 10, 12, 31 | hdmapinvlem2 37987 | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 ) |
56 | 55 | oveq2d 6921 | . . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 )) |
57 | 7, 9, 45 | ringrz 18942 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × 0 ) = 0 ) |
58 | 41, 14, 57 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (𝑛 × 0 ) = 0 ) |
59 | 56, 58 | eqtrd 2861 | . . . . 5 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = 0 ) |
60 | 59 | oveq1d 6920 | . . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦))) |
61 | 1, 2, 3, 6, 7, 10, 12, 28, 32 | hdmapipcl 37973 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) |
62 | 7, 8, 45 | grplid 17806 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦)) |
63 | 49, 61, 62 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦)) |
64 | 54, 60, 63 | 3eqtrd 2865 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑆‘𝑥)‘𝑦)) |
65 | 53, 64 | oveq12d 6923 | . 2 ⊢ (𝜑 → ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦))) |
66 | 34, 65 | eqtrd 2861 | 1 ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 {csn 4397 〈cop 4403 I cid 5249 ↾ cres 5344 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 .rcmulr 16306 Scalarcsca 16308 ·𝑠 cvsca 16309 0gc0g 16453 Grpcgrp 17776 LSSumclsm 18400 1rcur 18855 Ringcrg 18901 LModclmod 19219 LSpanclspn 19330 HLchlt 35418 LHypclh 36052 LTrncltrn 36169 DVecHcdvh 37146 ocHcoch 37415 HVMapchvm 37824 HDMapchdma 37860 HGMapchg 37951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35021 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-mre 16599 df-mrc 16600 df-acs 16602 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-oppg 18126 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35044 df-lshyp 35045 df-lcv 35087 df-lfl 35126 df-lkr 35154 df-ldual 35192 df-oposet 35244 df-ol 35246 df-oml 35247 df-covers 35334 df-ats 35335 df-atl 35366 df-cvlat 35390 df-hlat 35419 df-llines 35566 df-lplanes 35567 df-lvols 35568 df-lines 35569 df-psubsp 35571 df-pmap 35572 df-padd 35864 df-lhyp 36056 df-laut 36057 df-ldil 36172 df-ltrn 36173 df-trl 36227 df-tgrp 36811 df-tendo 36823 df-edring 36825 df-dveca 37071 df-disoa 37097 df-dvech 37147 df-dib 37207 df-dic 37241 df-dih 37297 df-doch 37416 df-djh 37463 df-lcdual 37655 df-mapd 37693 df-hvmap 37825 df-hdmap1 37861 df-hdmap 37862 df-hgmap 37952 |
This theorem is referenced by: hdmapglem7 37997 |
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