Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 38898. (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 38902 | . . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
13 | 12 | fveq1d 6665 | . 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 7107 | . . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V | |
16 | fvoveq1 7168 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣))) | |
17 | 16 | eqeq1d 2820 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
18 | 17 | ralbidv 3194 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
19 | 18 | riotabidv 7105 | . . . 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 2818 | . . . 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 6759 | . . 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 586 | . 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 2853 | 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 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ↦ cmpt 5137 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 LHypclh 37000 DVecHcdvh 38094 LCDualclcd 38602 HDMapchdma 38808 HGMapchg 38899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-hgmap 38900 |
This theorem is referenced by: hgmapcl 38905 hgmapvs 38907 |
Copyright terms: Public domain | W3C validator |