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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapval | Structured version Visualization version GIF version |
Description: Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 36192. (Contributed by NM, 25-Mar-2015.) |
Ref | Expression |
---|---|
hgmapval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmapfval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmapfval.v | ⊢ 𝑉 = (Base‘𝑈) |
hgmapfval.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hgmapfval.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmapfval.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmapfval.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hgmapfval.s | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hgmapfval.m | ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) |
hgmapfval.i | ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) |
hgmapfval.k | ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) |
hgmapval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
hgmapval | ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hgmapfval.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hgmapfval.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hgmapfval.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hgmapfval.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hgmapfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
7 | hgmapfval.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hgmapfval.s | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
9 | hgmapfval.m | . . . 4 ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) | |
10 | hgmapfval.i | . . . 4 ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) | |
11 | hgmapfval.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | hgmapfval 36196 | . . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
13 | 12 | fveq1d 6105 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋)) |
14 | hgmapval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | riotaex 6515 | . . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V | |
16 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑣) = (𝑋 · 𝑣)) | |
17 | 16 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣))) |
18 | 17 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
19 | 18 | ralbidv 2969 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
20 | 19 | riotabidv 6513 | . . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
21 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) | |
22 | 20, 21 | fvmptg 6189 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
23 | 14, 15, 22 | sylancl 693 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
24 | 13, 23 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 LHypclh 34288 DVecHcdvh 35385 LCDualclcd 35893 HDMapchdma 36100 HGMapchg 36193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-hgmap 36194 |
This theorem is referenced by: hgmapcl 36199 hgmapvs 36201 |
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