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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 38899 through hdmaplem4 38902, 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 2820 | . . 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 38958 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) |
15 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | hdmapval2lem 38959 | . . 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 234 | . 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 2898 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) | |
20 | 19 | notbid 320 | . . . 4 ⊢ (𝑧 = 𝑋 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ↔ ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) |
21 | oteq1 4804 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) | |
22 | oteq3 4806 | . . . . . . . . 9 ⊢ (𝑧 = 𝑋 → 〈𝐸, (𝐽‘𝐸), 𝑧〉 = 〈𝐸, (𝐽‘𝐸), 𝑋〉) | |
23 | 22 | fveq2d 6667 | . . . . . . . 8 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉)) |
24 | 23 | oteq2d 4808 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
25 | 21, 24 | eqtrd 2854 | . . . . . 6 ⊢ (𝑧 = 𝑋 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉 = 〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉) |
26 | 25 | fveq2d 6667 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) |
27 | 26 | eqeq2d 2830 | . . . 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 347 | . . 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 3618 | . 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 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ∪ cun 3932 {csn 4559 〈cop 4565 〈cotp 4567 I cid 5452 ↾ cres 5550 ‘cfv 6348 Basecbs 16475 LSpanclspn 19735 HLchlt 36478 LHypclh 37112 LTrncltrn 37229 DVecHcdvh 38206 LCDualclcd 38714 HVMapchvm 38884 HDMap1chdma1 38919 HDMapchdma 38920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-lshyp 36105 df-lcv 36147 df-lfl 36186 df-lkr 36214 df-ldual 36252 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tgrp 37871 df-tendo 37883 df-edring 37885 df-dveca 38131 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 df-djh 38523 df-lcdual 38715 df-mapd 38753 df-hvmap 38885 df-hdmap1 38921 df-hdmap 38922 |
This theorem is referenced by: hdmapval0 38961 hdmapeveclem 38962 hdmapval3lemN 38965 hdmap10lem 38967 hdmap11lem1 38969 |
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