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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 7097 | . . 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 2798 | . . 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 6463 | . 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 2798 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
7 | eqid 2798 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
8 | hgmapfn.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
9 | hgmapfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2798 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
11 | eqid 2798 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
12 | eqid 2798 | . . . 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 39182 | . . 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 6416 | . 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 261 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 Fn wfn 6319 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 HDMapchdma 39088 HGMapchg 39179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-hgmap 39180 |
This theorem is referenced by: hgmaprnlem1N 39192 hgmaprnN 39197 hgmapf1oN 39199 |
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