Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem7a | Structured version Visualization version GIF version |
Description: Lemma for hdmapg 40191. (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 39371 | . . . . . . 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 39373 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
15 | 14 | eldifad 3909 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
16 | hdmapglem7.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
17 | 5, 6, 16 | lspsncl 20337 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
18 | 9, 15, 17 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) |
19 | 15 | snssd 4755 | . . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
20 | 2, 4, 3, 5, 16, 8, 19 | dochocsp 39640 | . . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸})) |
21 | 20 | fveq2d 6823 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸}))) |
22 | 2, 4, 3, 5, 16, 8, 15 | dochocsn 39642 | . . . . . . 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 39729 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉) |
25 | 20 | oveq2d 7345 | . . . . 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 2839 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) |
28 | 6 | lsssssubg 20318 | . . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
29 | 9, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
30 | 29, 18 | sseldd 3932 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈)) |
31 | 2, 4, 5, 6, 3 | dochlss 39615 | . . . . . 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 3932 | . . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) |
34 | hdmapglem7.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
35 | 34, 7 | lsmelval 19342 | . . . 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 3269 | . . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢)) | |
39 | df-rex 3071 | . . . . 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 20363 | . . . . . . . . . 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 3181 | . . . . . . . 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 1923 | . . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) |
49 | rexcom4 3267 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
50 | ovex 7362 | . . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V | |
51 | oveq1 7336 | . . . . . . . . . 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 3488 | . . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) |
54 | 53 | rexbii 3093 | . . . . . . 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 3171 | . . 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 1540 ∃wex 1780 ∈ wcel 2105 ∃wrex 3070 ⊆ wss 3897 {csn 4572 〈cop 4578 I cid 5511 ↾ cres 5616 ‘cfv 6473 (class class class)co 7329 Basecbs 17001 +gcplusg 17051 Scalarcsca 17054 ·𝑠 cvsca 17055 0gc0g 17239 SubGrpcsubg 18837 LSSumclsm 19327 LModclmod 20221 LSubSpclss 20291 LSpanclspn 20331 HLchlt 37610 LHypclh 38245 LTrncltrn 38362 DVecHcdvh 39339 ocHcoch 39608 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-mre 17384 df-mrc 17385 df-acs 17387 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-oppg 19038 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lsatoms 37236 df-lcv 37279 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tgrp 39004 df-tendo 39016 df-edring 39018 df-dveca 39264 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609 df-djh 39656 |
This theorem is referenced by: hdmapglem7 40190 |
Copyright terms: Public domain | W3C validator |