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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfl0sc | Structured version Visualization version GIF version |
Description: The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of (𝑉 × { 0 }) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.) |
Ref | Expression |
---|---|
lfl0sc.v | ⊢ 𝑉 = (Base‘𝑊) |
lfl0sc.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lfl0sc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lfl0sc.k | ⊢ 𝐾 = (Base‘𝐷) |
lfl0sc.t | ⊢ · = (.r‘𝐷) |
lfl0sc.o | ⊢ 0 = (0g‘𝐷) |
lfl0sc.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lfl0sc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lfl0sc | ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × { 0 })) = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfl0sc.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6677 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfl0sc.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lfl0sc.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
6 | lfl0sc.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
7 | lfl0sc.k | . . . 4 ⊢ 𝐾 = (Base‘𝐷) | |
8 | lfl0sc.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 36079 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | 6 | lmodring 19571 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring) |
12 | 4, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Ring) |
13 | lfl0sc.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
14 | 7, 13 | ring0cl 19248 | . . 3 ⊢ (𝐷 ∈ Ring → 0 ∈ 𝐾) |
15 | 12, 14 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ 𝐾) |
16 | lfl0sc.t | . . . 4 ⊢ · = (.r‘𝐷) | |
17 | 7, 16, 13 | ringrz 19267 | . . 3 ⊢ ((𝐷 ∈ Ring ∧ 𝑘 ∈ 𝐾) → (𝑘 · 0 ) = 0 ) |
18 | 12, 17 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝑘 · 0 ) = 0 ) |
19 | 3, 10, 15, 15, 18 | caofid1 7428 | 1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × { 0 })) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 0gc0g 16701 Ringcrg 19226 LModclmod 19563 LFnlclfn 36073 |
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-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 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-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-mgp 19169 df-ring 19228 df-lmod 19565 df-lfl 36074 |
This theorem is referenced by: lkrscss 36114 lfl1dim 36137 lfl1dim2N 36138 |
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