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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapval | Structured version Visualization version GIF version |
Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector 𝐸 to be 〈0, 1〉 (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom 𝑃 = ((oc‘𝐾)‘𝑊) (see dvheveccl 37182). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 37839 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the ∀𝑧 ∈ 𝑉 ranges over. The middle term (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 37900. If a separate auxiliary vector is known, hdmapval2 37902 provides a version without quantification. (Contributed by NM, 15-May-2015.) |
Ref | Expression |
---|---|
hdmapval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapfval.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapfval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapfval.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapfval.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmapfval.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapfval.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapfval.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmapfval.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmapfval.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapfval.k | ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) |
hdmapval.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapval | ⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapfval.e | . . . 4 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
3 | hdmapfval.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hdmapfval.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
5 | hdmapfval.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | hdmapfval.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | hdmapfval.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
8 | hdmapfval.j | . . . 4 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
9 | hdmapfval.i | . . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
10 | hdmapfval.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hdmapfval.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | hdmapfval 37897 | . . 3 ⊢ (𝜑 → 𝑆 = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))))) |
13 | 12 | fveq1d 6439 | . 2 ⊢ (𝜑 → (𝑆‘𝑇) = ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))))‘𝑇)) |
14 | hdmapval.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
15 | riotaex 6875 | . . 3 ⊢ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) ∈ V | |
16 | sneq 4409 | . . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → {𝑡} = {𝑇}) | |
17 | 16 | fveq2d 6441 | . . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑁‘{𝑡}) = (𝑁‘{𝑇})) |
18 | 17 | uneq2d 3996 | . . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
19 | 18 | eleq2d 2892 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) |
20 | 19 | notbid 310 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))) |
21 | oteq3 4636 | . . . . . . . . 9 ⊢ (𝑡 = 𝑇 → 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉 = 〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) | |
22 | 21 | fveq2d 6441 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)) |
23 | 22 | eqeq2d 2835 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉) ↔ 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) |
24 | 20, 23 | imbi12d 336 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
25 | 24 | ralbidv 3195 | . . . . 5 ⊢ (𝑡 = 𝑇 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
26 | 25 | riotabidv 6873 | . . . 4 ⊢ (𝑡 = 𝑇 → (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
27 | eqid 2825 | . . . 4 ⊢ (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉)))) = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉)))) | |
28 | 26, 27 | fvmptg 6531 | . . 3 ⊢ ((𝑇 ∈ 𝑉 ∧ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) ∈ V) → ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))))‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
29 | 14, 15, 28 | sylancl 580 | . 2 ⊢ (𝜑 → ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))))‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
30 | 13, 29 | eqtrd 2861 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ∪ cun 3796 {csn 4399 〈cop 4405 〈cotp 4407 ↦ cmpt 4954 I cid 5251 ↾ cres 5348 ‘cfv 6127 ℩crio 6870 Basecbs 16229 LSpanclspn 19337 LHypclh 36054 LTrncltrn 36171 DVecHcdvh 37148 LCDualclcd 37656 HVMapchvm 37826 HDMap1chdma1 37861 HDMapchdma 37862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-ot 4408 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-hdmap 37864 |
This theorem is referenced by: hdmapcl 37900 hdmapval2lem 37901 |
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