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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 39020. (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 39024 | . . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
13 | 12 | fveq1d 6674 | . 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 7120 | . . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V | |
16 | fvoveq1 7181 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣))) | |
17 | 16 | eqeq1d 2825 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
18 | 17 | ralbidv 3199 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
19 | 18 | riotabidv 7118 | . . . 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 2823 | . . . 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 6768 | . . 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 588 | . 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 2858 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 HDMapchdma 38930 HGMapchg 39021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-hgmap 39022 |
This theorem is referenced by: hgmapcl 39027 hgmapvs 39029 |
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