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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1eu | Structured version Visualization version GIF version |
Description: Convert mapdh9a 41748 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) |
Ref | Expression |
---|---|
hdmap1eu.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1eu.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1eu.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1eu.o | ⊢ 0 = (0g‘𝑈) |
hdmap1eu.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1eu.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1eu.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1eu.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1eu.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1eu.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1eu.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1eu.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
hdmap1eu.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1eu.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1eu.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmap1eu | ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1eu.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1eu.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1eu.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | eqid 2740 | . 2 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
5 | hdmap1eu.o | . 2 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap1eu.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap1eu.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap1eu.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
9 | eqid 2740 | . 2 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
10 | eqid 2740 | . 2 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
11 | hdmap1eu.l | . 2 ⊢ 𝐿 = (LSpan‘𝐶) | |
12 | hdmap1eu.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | hdmap1eu.i | . 2 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
14 | hdmap1eu.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | hdmap1eu.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
16 | hdmap1eu.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
17 | hdmap1eu.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1eu.t | . 2 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
19 | eqid 2740 | . . 3 ⊢ (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)}))))) = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)}))))) | |
20 | 19 | hdmap1cbv 41761 | . 2 ⊢ (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)}))))) = (𝑤 ∈ V ↦ if((2nd ‘𝑤) = 0 , (0g‘𝐶), (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑤)})) = (𝐿‘{𝑔}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑤))(-g‘𝑈)(2nd ‘𝑤))})) = (𝐿‘{((2nd ‘(1st ‘𝑤))(-g‘𝐶)𝑔)}))))) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20 | hdmap1eulem 41781 | 1 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃!wreu 3386 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ifcif 4548 {csn 4648 〈cotp 4656 ↦ cmpt 5249 ‘cfv 6575 ℩crio 7405 (class class class)co 7450 1st c1st 8030 2nd c2nd 8031 Basecbs 17260 0gc0g 17501 -gcsg 18977 LSpanclspn 20994 HLchlt 39308 LHypclh 39943 DVecHcdvh 41037 LCDualclcd 41545 mapdcmpd 41583 HDMap1chdma1 41750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-riotaBAD 38911 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-undef 8316 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-0g 17503 df-mre 17646 df-mrc 17647 df-acs 17649 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19359 df-oppg 19388 df-lsm 19680 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20541 df-rlreg 20718 df-domn 20719 df-drng 20755 df-lmod 20884 df-lss 20955 df-lsp 20995 df-lvec 21127 df-lsatoms 38934 df-lshyp 38935 df-lcv 38977 df-lfl 39016 df-lkr 39044 df-ldual 39082 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118 df-tgrp 40702 df-tendo 40714 df-edring 40716 df-dveca 40962 df-disoa 40988 df-dvech 41038 df-dib 41098 df-dic 41132 df-dih 41188 df-doch 41307 df-djh 41354 df-lcdual 41546 df-mapd 41584 df-hdmap1 41752 |
This theorem is referenced by: hdmapcl 41789 hdmapval2lem 41790 |
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