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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 7392 | . . 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 2735 | . . 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 6712 | . 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 2735 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
7 | eqid 2735 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
8 | hgmapfn.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
9 | hgmapfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
10 | eqid 2735 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
11 | eqid 2735 | . . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) | |
12 | eqid 2735 | . . . 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 41869 | . . 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 6662 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ↦ cmpt 5231 Fn wfn 6558 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 HDMapchdma 41775 HGMapchg 41866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-hgmap 41867 |
This theorem is referenced by: hgmaprnlem1N 41879 hgmaprnN 41884 hgmapf1oN 41886 |
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