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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 4596 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → {𝑇} = {(0g‘𝑈)}) | |
2 | 1 | fveq2d 6846 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → (𝑁‘{𝑇}) = (𝑁‘{(0g‘𝑈)})) |
3 | 2 | fveq2d 6846 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝑀‘(𝑁‘{𝑇})) = (𝑀‘(𝑁‘{(0g‘𝑈)}))) |
4 | fveq2 6842 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → (𝑆‘𝑇) = (𝑆‘(0g‘𝑈))) | |
5 | 4 | sneqd 4598 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → {(𝑆‘𝑇)} = {(𝑆‘(0g‘𝑈))}) |
6 | 5 | fveq2d 6846 | . . 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 4747 | . . . 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 40302 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
28 | 8, 9, 16 | dvhlmod 39573 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | 19, 11 | lspsn0 20469 | . . . . 5 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
31 | 30 | fveq2d 6846 | . . 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 40128 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
34 | 8, 9, 19, 12, 32, 15, 16 | hdmapval0 40296 | . . . . . 6 ⊢ (𝜑 → (𝑆‘(0g‘𝑈)) = (0g‘𝐶)) |
35 | 34 | sneqd 4598 | . . . . 5 ⊢ (𝜑 → {(𝑆‘(0g‘𝑈))} = {(0g‘𝐶)}) |
36 | 35 | fveq2d 6846 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘(0g‘𝑈))}) = (𝐿‘{(0g‘𝐶)})) |
37 | 8, 12, 16 | lcdlmod 40055 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
38 | 32, 13 | lspsn0 20469 | . . . . 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 3030 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 〈cop 4592 I cid 5530 ↾ cres 5635 ‘cfv 6496 Basecbs 17083 0gc0g 17321 LModclmod 20322 LSpanclspn 20432 HLchlt 37812 LHypclh 38447 LTrncltrn 38564 DVecHcdvh 39541 LCDualclcd 40049 mapdcmpd 40087 HVMapchvm 40219 HDMap1chdma1 40254 HDMapchdma 40255 |
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-ot 4595 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-of 7617 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-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 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 df-lcdual 40050 df-mapd 40088 df-hvmap 40220 df-hdmap1 40256 df-hdmap 40257 |
This theorem is referenced by: hdmapeq0 40307 hdmaprnlem1N 40312 hdmaprnlem3uN 40314 hdmaprnlem6N 40317 hdmaprnlem8N 40319 hdmaprnlem3eN 40321 hdmap14lem1a 40329 hdmap14lem9 40339 hgmaprnlem2N 40360 hdmaplkr 40376 |
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