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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 7393 | . . 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 2736 | . . 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 6710 | . 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 2736 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 7 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
| 8 | hgmapfn.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 9 | hgmapfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 10 | eqid 2736 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
| 11 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
| 12 | eqid 2736 | . . . 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 41889 | . . 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 6660 | . 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 258 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ↦ cmpt 5224 Fn wfn 6555 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 HLchlt 39352 LHypclh 39987 DVecHcdvh 41081 LCDualclcd 41589 HDMapchdma 41795 HGMapchg 41886 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-hgmap 41887 | 
| This theorem is referenced by: hgmaprnlem1N 41899 hgmaprnN 41904 hgmapf1oN 41906 | 
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