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Mathbox for Norm Megill |
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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 37956. (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 37960 | . . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
13 | 12 | fveq1d 6439 | . 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 6875 | . . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V | |
16 | fvoveq1 6933 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣))) | |
17 | 16 | eqeq1d 2827 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
18 | 17 | ralbidv 3195 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
19 | 18 | riotabidv 6873 | . . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
20 | eqid 2825 | . . . 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)))) | |
21 | 19, 20 | fvmptg 6531 | . . 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)))) |
22 | 14, 15, 21 | sylancl 580 | . 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)))) |
23 | 13, 22 | eqtrd 2861 | 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 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ↦ cmpt 4954 ‘cfv 6127 ℩crio 6870 (class class class)co 6910 Basecbs 16229 Scalarcsca 16315 ·𝑠 cvsca 16316 LHypclh 36058 DVecHcdvh 37152 LCDualclcd 37660 HDMapchdma 37866 HGMapchg 37957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-hgmap 37958 |
This theorem is referenced by: hgmapcl 37963 hgmapvs 37965 |
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