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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapval2 | Structured version Visualization version GIF version |
Description: Value of map from vectors to functionals with a specific auxiliary vector. TODO: Would shorter proofs result if the .ne hypothesis were changed to two ≠ hypothesis? Consider hdmaplem1 37377 through hdmaplem4 37380, which would become obsolete. (Contributed by NM, 15-May-2015.) |
Ref | Expression |
---|---|
hdmapval2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapval2.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapval2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapval2.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapval2.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmapval2.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapval2.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapval2.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmapval2.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmapval2.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapval2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapval2.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
hdmapval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmapval2.ne | ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
Ref | Expression |
---|---|
hdmapval2 | ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2652 | . . 3 ⊢ (𝜑 → (𝑆‘𝑇) = (𝑆‘𝑇)) | |
2 | hdmapval2.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | hdmapval2.e | . . . 4 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
4 | hdmapval2.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | hdmapval2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
6 | hdmapval2.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmapval2.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmapval2.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmapval2.j | . . . 4 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
10 | hdmapval2.i | . . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
11 | hdmapval2.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
12 | hdmapval2.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | hdmapval2.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
14 | 2, 4, 5, 7, 8, 11, 12, 13 | hdmapcl 37439 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) |
15 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | hdmapval2lem 37440 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑇) = (𝑆‘𝑇) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
16 | 1, 15 | mpbid 222 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) |
17 | hdmapval2.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
18 | hdmapval2.ne | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) | |
19 | eleq1 2718 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) | |
20 | 19 | notbid 307 | . . . 4 ⊢ (𝑧 = 𝑋 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) |
21 | oteq1 4442 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) | |
22 | oteq3 4444 | . . . . . . . . 9 ⊢ (𝑧 = 𝑋 → 〈𝐸, (𝐽‘𝐸), 𝑧〉 = 〈𝐸, (𝐽‘𝐸), 𝑋〉) | |
23 | 22 | fveq2d 6233 | . . . . . . . 8 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉)) |
24 | 23 | oteq2d 4446 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
25 | 21, 24 | eqtrd 2685 | . . . . . 6 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
26 | 25 | fveq2d 6233 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) |
27 | 26 | eqeq2d 2661 | . . . 4 ⊢ (𝑧 = 𝑋 → ((𝑆‘𝑇) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) ↔ (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉))) |
28 | 20, 27 | imbi12d 333 | . . 3 ⊢ (𝑧 = 𝑋 → ((¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)) ↔ (¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)))) |
29 | 28 | rspccv 3337 | . 2 ⊢ (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)) → (𝑋 ∈ 𝑉 → (¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)))) |
30 | 16, 17, 18, 29 | syl3c 66 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∪ cun 3605 {csn 4210 〈cop 4216 〈cotp 4218 I cid 5052 ↾ cres 5145 ‘cfv 5926 Basecbs 15904 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 LTrncltrn 35705 DVecHcdvh 36684 LCDualclcd 37192 HVMapchvm 37362 HDMap1chdma1 37398 HDMapchdma 37399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 df-hvmap 37363 df-hdmap1 37400 df-hdmap 37401 |
This theorem is referenced by: hdmapval0 37442 hdmapeveclem 37443 hdmapval3lemN 37446 hdmap10lem 37448 hdmap11lem1 37450 |
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