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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem7a | Structured version Visualization version GIF version |
Description: Lemma for hdmapg 41887. (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) |
Ref | Expression |
---|---|
hdmapglem7a | ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglem7.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | hdmapglem7.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | hdmapglem7.o | . . . . . 6 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
4 | hdmapglem7.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | hdmapglem7.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
6 | eqid 2740 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
7 | hdmapglem7.a | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑈) | |
8 | hdmapglem7.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 2, 4, 8 | dvhlmod 41067 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | eqid 2740 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | eqid 2740 | . . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
12 | eqid 2740 | . . . . . . . . 9 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
13 | hdmapglem7.e | . . . . . . . . 9 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
14 | 2, 10, 11, 4, 5, 12, 13, 8 | dvheveccl 41069 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
15 | 14 | eldifad 3988 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
16 | hdmapglem7.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
17 | 5, 6, 16 | lspsncl 20998 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
18 | 9, 15, 17 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
19 | 15 | snssd 4834 | . . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
20 | 2, 4, 3, 5, 16, 8, 19 | dochocsp 41336 | . . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸})) |
21 | 20 | fveq2d 6924 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸}))) |
22 | 2, 4, 3, 5, 16, 8, 15 | dochocsn 41338 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸})) |
23 | 21, 22 | eqtrd 2780 | . . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸})) |
24 | 2, 3, 4, 5, 6, 7, 8, 18, 23 | dochexmid 41425 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉) |
25 | 20 | oveq2d 7464 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
26 | 24, 25 | eqtr3d 2782 | . . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
27 | 1, 26 | eleqtrd 2846 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
28 | 6 | lsssssubg 20979 | . . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
29 | 9, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
30 | 29, 18 | sseldd 4009 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈)) |
31 | 2, 4, 5, 6, 3 | dochlss 41311 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) |
32 | 8, 19, 31 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) |
33 | 29, 32 | sseldd 4009 | . . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) |
34 | hdmapglem7.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
35 | 34, 7 | lsmelval 19691 | . . . 4 ⊢ (((𝑁‘{𝐸}) ∈ (SubGrp‘𝑈) ∧ (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))) |
36 | 30, 33, 35 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))) |
37 | 27, 36 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)) |
38 | rexcom 3296 | . . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢)) | |
39 | df-rex 3077 | . . . . 5 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢))) | |
40 | hdmapglem7.r | . . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑈) | |
41 | hdmapglem7.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅) | |
42 | hdmapglem7.q | . . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑈) | |
43 | 40, 41, 5, 42, 16 | ellspsn 21024 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸))) |
44 | 9, 15, 43 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸))) |
45 | 44 | anbi1d 630 | . . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
46 | r19.41v 3195 | . . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
47 | 45, 46 | bitr4di 289 | . . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
48 | 47 | exbidv 1920 | . . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
49 | rexcom4 3294 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
50 | ovex 7481 | . . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V | |
51 | oveq1 7455 | . . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢)) | |
52 | 51 | eqeq2d 2751 | . . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
53 | 50, 52 | ceqsexv 3542 | . . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
54 | 53 | rexbii 3100 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
55 | 49, 54 | bitr3i 277 | . . . . . 6 ⊢ (∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
56 | 48, 55 | bitrdi 287 | . . . . 5 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
57 | 39, 56 | bitrid 283 | . . . 4 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
58 | 57 | rexbidv 3185 | . . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
59 | 38, 58 | bitrid 283 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
60 | 37, 59 | mpbid 232 | 1 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∃wrex 3076 ⊆ wss 3976 {csn 4648 〈cop 4654 I cid 5592 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +gcplusg 17311 Scalarcsca 17314 ·𝑠 cvsca 17315 0gc0g 17499 SubGrpcsubg 19160 LSSumclsm 19676 LModclmod 20880 LSubSpclss 20952 LSpanclspn 20992 HLchlt 39306 LHypclh 39941 LTrncltrn 40058 DVecHcdvh 41035 ocHcoch 41304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-mre 17644 df-mrc 17645 df-acs 17647 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-lcv 38975 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tgrp 40700 df-tendo 40712 df-edring 40714 df-dveca 40960 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305 df-djh 41352 |
This theorem is referenced by: hdmapglem7 41886 |
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