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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldual0v | Structured version Visualization version GIF version |
Description: The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.) |
Ref | Expression |
---|---|
ldualv0.v | ⊢ 𝑉 = (Base‘𝑊) |
ldualv0.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualv0.z | ⊢ 0 = (0g‘𝑅) |
ldualv0.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualv0.o | ⊢ 𝑂 = (0g‘𝐷) |
ldualv0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
Ref | Expression |
---|---|
ldual0v | ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
2 | ldualv0.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | ldualv0.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | eqid 2818 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
6 | ldualv0.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | ldualv0.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
8 | ldualv0.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
9 | 2, 7, 8, 1 | lfl0f 36085 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
11 | 1, 2, 3, 4, 5, 6, 10, 10 | ldualvadd 36145 | . . 3 ⊢ (𝜑 → ((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = ((𝑉 × { 0 }) ∘f (+g‘𝑅)(𝑉 × { 0 }))) |
12 | 8, 2, 3, 7, 1, 6, 10 | lfladd0l 36090 | . . 3 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f (+g‘𝑅)(𝑉 × { 0 })) = (𝑉 × { 0 })) |
13 | 11, 12 | eqtrd 2853 | . 2 ⊢ (𝜑 → ((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 })) |
14 | 4, 6 | ldualgrp 36162 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Grp) |
15 | eqid 2818 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | 1, 4, 15, 6, 10 | ldualelvbase 36143 | . . 3 ⊢ (𝜑 → (𝑉 × { 0 }) ∈ (Base‘𝐷)) |
17 | ldualv0.o | . . . 4 ⊢ 𝑂 = (0g‘𝐷) | |
18 | 15, 5, 17 | grpid 18077 | . . 3 ⊢ ((𝐷 ∈ Grp ∧ (𝑉 × { 0 }) ∈ (Base‘𝐷)) → (((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 }) ↔ 𝑂 = (𝑉 × { 0 }))) |
19 | 14, 16, 18 | syl2anc 584 | . 2 ⊢ (𝜑 → (((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 }) ↔ 𝑂 = (𝑉 × { 0 }))) |
20 | 13, 19 | mpbid 233 | 1 ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 0gc0g 16701 Grpcgrp 18041 LModclmod 19563 LFnlclfn 36073 LDualcld 36139 |
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-int 4868 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-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 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-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-sca 16569 df-vsca 16570 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 df-lfl 36074 df-ldual 36140 |
This theorem is referenced by: ldual0vcl 36167 lkr0f2 36177 lduallkr3 36178 lclkrlem1 38522 lclkrlem2j 38532 lcd0v 38627 lcd0v2 38628 |
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