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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem7a | Structured version Visualization version GIF version |
Description: Lemma for hdmapg 40393. (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 2736 | . . . . . 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 39573 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | eqid 2736 | . . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
12 | eqid 2736 | . . . . . . . . 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 39575 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
15 | 14 | eldifad 3922 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
16 | hdmapglem7.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
17 | 5, 6, 16 | lspsncl 20438 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
18 | 9, 15, 17 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
19 | 15 | snssd 4769 | . . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
20 | 2, 4, 3, 5, 16, 8, 19 | dochocsp 39842 | . . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸})) |
21 | 20 | fveq2d 6846 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸}))) |
22 | 2, 4, 3, 5, 16, 8, 15 | dochocsn 39844 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸})) |
23 | 21, 22 | eqtrd 2776 | . . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸})) |
24 | 2, 3, 4, 5, 6, 7, 8, 18, 23 | dochexmid 39931 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉) |
25 | 20 | oveq2d 7373 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
26 | 24, 25 | eqtr3d 2778 | . . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
27 | 1, 26 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
28 | 6 | lsssssubg 20419 | . . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
29 | 9, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
30 | 29, 18 | sseldd 3945 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈)) |
31 | 2, 4, 5, 6, 3 | dochlss 39817 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) |
32 | 8, 19, 31 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) |
33 | 29, 32 | sseldd 3945 | . . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) |
34 | hdmapglem7.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
35 | 34, 7 | lsmelval 19431 | . . . 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 584 | . . 3 ⊢ (𝜑 → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))) |
37 | 27, 36 | mpbid 231 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)) |
38 | rexcom 3273 | . . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢)) | |
39 | df-rex 3074 | . . . . 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 | lspsnel 20464 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸))) |
44 | 9, 15, 43 | syl2anc 584 | . . . . . . . . 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 3185 | . . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
47 | 45, 46 | bitr4di 288 | . . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
48 | 47 | exbidv 1924 | . . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
49 | rexcom4 3271 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
50 | ovex 7390 | . . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V | |
51 | oveq1 7364 | . . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢)) | |
52 | 51 | eqeq2d 2747 | . . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
53 | 50, 52 | ceqsexv 3494 | . . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
54 | 53 | rexbii 3097 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
55 | 49, 54 | bitr3i 276 | . . . . . 6 ⊢ (∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
56 | 48, 55 | bitrdi 286 | . . . . 5 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
57 | 39, 56 | bitrid 282 | . . . 4 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
58 | 57 | rexbidv 3175 | . . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
59 | 38, 58 | bitrid 282 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) |
60 | 37, 59 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3073 ⊆ wss 3910 {csn 4586 〈cop 4592 I cid 5530 ↾ cres 5635 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 SubGrpcsubg 18922 LSSumclsm 19416 LModclmod 20322 LSubSpclss 20392 LSpanclspn 20432 HLchlt 37812 LHypclh 38447 LTrncltrn 38564 DVecHcdvh 39541 ocHcoch 39810 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lcv 37481 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 |
This theorem is referenced by: hdmapglem7 40392 |
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