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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapeveclem | Structured version Visualization version GIF version |
Description: Lemma for hdmapevec 38965. TODO: combine with hdmapevec 38965 if it shortens overall. (Contributed by NM, 16-May-2015.) |
Ref | Expression |
---|---|
hdmapevec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapevec.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapevec.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmapevec.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapevec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapevec.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapevec.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapevec.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmapevec.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapevec.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapevec.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmapevec.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmapevec.ne | ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) |
Ref | Expression |
---|---|
hdmapeveclem | ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapevec.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapevec.e | . . 3 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
3 | hdmapevec.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hdmapevec.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | hdmapevec.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | hdmapevec.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | hdmapevec.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
8 | hdmapevec.j | . . 3 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
9 | hdmapevec.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
10 | hdmapevec.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hdmapevec.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2821 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
14 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
15 | 1, 12, 13, 3, 4, 14, 2, 11 | dvheveccl 38242 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
16 | 15 | eldifad 3948 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
17 | hdmapevec.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
18 | hdmapevec.ne | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18 | hdmapval2 38962 | . 2 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝐸〉)) |
20 | eqid 2821 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
21 | eqid 2821 | . . 3 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
22 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
23 | 1, 3, 4, 14, 6, 7, 22, 8, 11, 15 | hvmapcl2 38896 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
24 | 23 | eldifad 3948 | . . 3 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
25 | 1, 3, 4, 14, 5, 6, 20, 21, 8, 11, 15 | mapdhvmap 38899 | . . 3 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
26 | 1, 3, 11 | dvhlmod 38240 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | 4, 5, 26, 17, 18, 16 | hdmaplem1 38901 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝐸})) |
28 | 27 | necomd 3071 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑋})) |
29 | 4, 5, 26, 17, 18, 16, 14 | hdmaplem3 38903 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
30 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉)) | |
31 | 1, 3, 4, 14, 5, 6, 7, 20, 21, 9, 11, 24, 25, 28, 15, 29, 30 | hdmap1eq2 38935 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝐸〉) = (𝐽‘𝐸)) |
32 | 19, 31 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 {csn 4561 〈cop 4567 〈cotp 4569 I cid 5454 ↾ cres 5552 ‘cfv 6350 Basecbs 16477 0gc0g 16707 LSpanclspn 19737 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 DVecHcdvh 38208 LCDualclcd 38716 mapdcmpd 38754 HVMapchvm 38886 HDMap1chdma1 38921 HDMapchdma 38922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lcv 36149 df-lfl 36188 df-lkr 36216 df-ldual 36254 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 df-djh 38525 df-lcdual 38717 df-mapd 38755 df-hvmap 38887 df-hdmap1 38923 df-hdmap 38924 |
This theorem is referenced by: hdmapevec 38965 |
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