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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapfnN | Structured version Visualization version GIF version |
Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hgmapfn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmapfn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmapfn.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmapfn.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmapfn.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmapfn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hgmapfnN | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaex 7379 | . . 3 ⊢ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V | |
2 | eqid 2725 | . . 3 ⊢ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) = (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) | |
3 | 1, 2 | fnmpti 6699 | . 2 ⊢ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵 |
4 | hgmapfn.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | hgmapfn.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2725 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
7 | eqid 2725 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
8 | hgmapfn.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
9 | hgmapfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2725 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
11 | eqid 2725 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
12 | eqid 2725 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
13 | hgmapfn.g | . . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
14 | hgmapfn.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | hgmapfval 41486 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))))) |
16 | 15 | fneq1d 6648 | . 2 ⊢ (𝜑 → (𝐺 Fn 𝐵 ↔ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵)) |
17 | 3, 16 | mpbiri 257 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ↦ cmpt 5232 Fn wfn 6544 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·𝑠 cvsca 17240 HLchlt 38949 LHypclh 39584 DVecHcdvh 40678 LCDualclcd 41186 HDMapchdma 41392 HGMapchg 41483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-hgmap 41484 |
This theorem is referenced by: hgmaprnlem1N 41496 hgmaprnN 41501 hgmapf1oN 41503 |
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