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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 39827 through hdmaplem4 39830, 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 2737 | . . 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 39886 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) |
15 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | hdmapval2lem 39887 | . . 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 231 | . 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 2824 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) | |
20 | 19 | notbid 318 | . . . 4 ⊢ (𝑧 = 𝑋 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) |
21 | oteq1 4818 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) | |
22 | oteq3 4820 | . . . . . . . . 9 ⊢ (𝑧 = 𝑋 → 〈𝐸, (𝐽‘𝐸), 𝑧〉 = 〈𝐸, (𝐽‘𝐸), 𝑋〉) | |
23 | 22 | fveq2d 6808 | . . . . . . . 8 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉)) |
24 | 23 | oteq2d 4822 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
25 | 21, 24 | eqtrd 2776 | . . . . . 6 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
26 | 25 | fveq2d 6808 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) |
27 | 26 | eqeq2d 2747 | . . . 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 345 | . . 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 3563 | . 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 397 = wceq 1539 ∈ wcel 2104 ∀wral 3062 ∪ cun 3890 {csn 4565 〈cop 4571 〈cotp 4573 I cid 5499 ↾ cres 5602 ‘cfv 6458 Basecbs 16957 LSpanclspn 20278 HLchlt 37406 LHypclh 38040 LTrncltrn 38157 DVecHcdvh 39134 LCDualclcd 39642 HVMapchvm 39812 HDMap1chdma1 39847 HDMapchdma 39848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-riotaBAD 37009 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-ot 4574 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-tpos 8073 df-undef 8120 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-0g 17197 df-mre 17340 df-mrc 17341 df-acs 17343 df-proset 18058 df-poset 18076 df-plt 18093 df-lub 18109 df-glb 18110 df-join 18111 df-meet 18112 df-p0 18188 df-p1 18189 df-lat 18195 df-clat 18262 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-subg 18797 df-cntz 18968 df-oppg 18995 df-lsm 19286 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-oppr 19907 df-dvdsr 19928 df-unit 19929 df-invr 19959 df-dvr 19970 df-drng 20038 df-lmod 20170 df-lss 20239 df-lsp 20279 df-lvec 20410 df-lsatoms 37032 df-lshyp 37033 df-lcv 37075 df-lfl 37114 df-lkr 37142 df-ldual 37180 df-oposet 37232 df-ol 37234 df-oml 37235 df-covers 37322 df-ats 37323 df-atl 37354 df-cvlat 37378 df-hlat 37407 df-llines 37554 df-lplanes 37555 df-lvols 37556 df-lines 37557 df-psubsp 37559 df-pmap 37560 df-padd 37852 df-lhyp 38044 df-laut 38045 df-ldil 38160 df-ltrn 38161 df-trl 38215 df-tgrp 38799 df-tendo 38811 df-edring 38813 df-dveca 39059 df-disoa 39085 df-dvech 39135 df-dib 39195 df-dic 39229 df-dih 39285 df-doch 39404 df-djh 39451 df-lcdual 39643 df-mapd 39681 df-hvmap 39813 df-hdmap1 39849 df-hdmap 39850 |
This theorem is referenced by: hdmapval0 39889 hdmapeveclem 39890 hdmapval3lemN 39893 hdmap10lem 39895 hdmap11lem1 39897 |
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