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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap10 | Structured version Visualization version GIF version |
Description: Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.) |
Ref | Expression |
---|---|
hdmap10.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap10.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap10.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap10.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap10.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap10.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap10.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap10.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap10.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap10.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmap10 | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4584 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → {𝑇} = {(0g‘𝑈)}) | |
2 | 1 | fveq2d 6830 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → (𝑁‘{𝑇}) = (𝑁‘{(0g‘𝑈)})) |
3 | 2 | fveq2d 6830 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝑀‘(𝑁‘{𝑇})) = (𝑀‘(𝑁‘{(0g‘𝑈)}))) |
4 | fveq2 6826 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → (𝑆‘𝑇) = (𝑆‘(0g‘𝑈))) | |
5 | 4 | sneqd 4586 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → {(𝑆‘𝑇)} = {(𝑆‘(0g‘𝑈))}) |
6 | 5 | fveq2d 6830 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝐿‘{(𝑆‘𝑇)}) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
7 | 3, 6 | eqeq12d 2752 | . 2 ⊢ (𝑇 = (0g‘𝑈) → ((𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)}) ↔ (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝐿‘{(𝑆‘(0g‘𝑈))}))) |
8 | hdmap10.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | hdmap10.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | hdmap10.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
11 | hdmap10.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
12 | hdmap10.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
13 | hdmap10.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap10.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap10.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap10.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
18 | eqid 2736 | . . 3 ⊢ 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | eqid 2736 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | eqid 2736 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
21 | eqid 2736 | . . 3 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊) | |
22 | eqid 2736 | . . 3 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊) | |
23 | hdmap10.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
24 | 23 | anim1i 615 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) |
25 | eldifsn 4735 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)}) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) | |
26 | 24, 25 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
27 | 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 26 | hdmap10lem 40158 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
28 | 8, 9, 16 | dvhlmod 39429 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | 19, 11 | lspsn0 20377 | . . . . 5 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
31 | 30 | fveq2d 6830 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝑀‘{(0g‘𝑈)})) |
32 | eqid 2736 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
33 | 8, 14, 9, 19, 12, 32, 16 | mapd0 39984 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
34 | 8, 9, 19, 12, 32, 15, 16 | hdmapval0 40152 | . . . . . 6 ⊢ (𝜑 → (𝑆‘(0g‘𝑈)) = (0g‘𝐶)) |
35 | 34 | sneqd 4586 | . . . . 5 ⊢ (𝜑 → {(𝑆‘(0g‘𝑈))} = {(0g‘𝐶)}) |
36 | 35 | fveq2d 6830 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘(0g‘𝑈))}) = (𝐿‘{(0g‘𝐶)})) |
37 | 8, 12, 16 | lcdlmod 39911 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
38 | 32, 13 | lspsn0 20377 | . . . . 5 ⊢ (𝐶 ∈ LMod → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
40 | 36, 39 | eqtr2d 2777 | . . 3 ⊢ (𝜑 → {(0g‘𝐶)} = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
41 | 31, 33, 40 | 3eqtrd 2780 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
42 | 7, 27, 41 | pm2.61ne 3027 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∖ cdif 3895 {csn 4574 〈cop 4580 I cid 5518 ↾ cres 5623 ‘cfv 6480 Basecbs 17010 0gc0g 17248 LModclmod 20230 LSpanclspn 20340 HLchlt 37668 LHypclh 38303 LTrncltrn 38420 DVecHcdvh 39397 LCDualclcd 39905 mapdcmpd 39943 HVMapchvm 40075 HDMap1chdma1 40110 HDMapchdma 40111 |
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 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-riotaBAD 37271 |
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 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-ot 4583 df-uni 4854 df-int 4896 df-iun 4944 df-iin 4945 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-of 7596 df-om 7782 df-1st 7900 df-2nd 7901 df-tpos 8113 df-undef 8160 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-er 8570 df-map 8689 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-nn 12076 df-2 12138 df-3 12139 df-4 12140 df-5 12141 df-6 12142 df-n0 12336 df-z 12422 df-uz 12685 df-fz 13342 df-struct 16946 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-mulr 17074 df-sca 17076 df-vsca 17077 df-0g 17250 df-mre 17393 df-mrc 17394 df-acs 17396 df-proset 18111 df-poset 18129 df-plt 18146 df-lub 18162 df-glb 18163 df-join 18164 df-meet 18165 df-p0 18241 df-p1 18242 df-lat 18248 df-clat 18315 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-submnd 18529 df-grp 18677 df-minusg 18678 df-sbg 18679 df-subg 18849 df-cntz 19020 df-oppg 19047 df-lsm 19338 df-cmn 19484 df-abl 19485 df-mgp 19817 df-ur 19834 df-ring 19881 df-oppr 19958 df-dvdsr 19979 df-unit 19980 df-invr 20010 df-dvr 20021 df-drng 20096 df-lmod 20232 df-lss 20301 df-lsp 20341 df-lvec 20472 df-lsatoms 37294 df-lshyp 37295 df-lcv 37337 df-lfl 37376 df-lkr 37404 df-ldual 37442 df-oposet 37494 df-ol 37496 df-oml 37497 df-covers 37584 df-ats 37585 df-atl 37616 df-cvlat 37640 df-hlat 37669 df-llines 37817 df-lplanes 37818 df-lvols 37819 df-lines 37820 df-psubsp 37822 df-pmap 37823 df-padd 38115 df-lhyp 38307 df-laut 38308 df-ldil 38423 df-ltrn 38424 df-trl 38478 df-tgrp 39062 df-tendo 39074 df-edring 39076 df-dveca 39322 df-disoa 39348 df-dvech 39398 df-dib 39458 df-dic 39492 df-dih 39548 df-doch 39667 df-djh 39714 df-lcdual 39906 df-mapd 39944 df-hvmap 40076 df-hdmap1 40112 df-hdmap 40113 |
This theorem is referenced by: hdmapeq0 40163 hdmaprnlem1N 40168 hdmaprnlem3uN 40170 hdmaprnlem6N 40173 hdmaprnlem8N 40175 hdmaprnlem3eN 40177 hdmap14lem1a 40185 hdmap14lem9 40195 hgmaprnlem2N 40216 hdmaplkr 40232 |
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