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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 2823 | . . . 4 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
2 | ldualv0.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | eqid 2823 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | ldualv0.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | eqid 2823 | . . . 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 36207 | . . . . 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 36267 | . . 3 ⊢ (𝜑 → ((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = ((𝑉 × { 0 }) ∘f (+g‘𝑅)(𝑉 × { 0 }))) |
12 | 8, 2, 3, 7, 1, 6, 10 | lfladd0l 36212 | . . 3 ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f (+g‘𝑅)(𝑉 × { 0 })) = (𝑉 × { 0 })) |
13 | 11, 12 | eqtrd 2858 | . 2 ⊢ (𝜑 → ((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 })) |
14 | 4, 6 | ldualgrp 36284 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Grp) |
15 | eqid 2823 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | 1, 4, 15, 6, 10 | ldualelvbase 36265 | . . 3 ⊢ (𝜑 → (𝑉 × { 0 }) ∈ (Base‘𝐷)) |
17 | ldualv0.o | . . . 4 ⊢ 𝑂 = (0g‘𝐷) | |
18 | 15, 5, 17 | grpid 18141 | . . 3 ⊢ ((𝐷 ∈ Grp ∧ (𝑉 × { 0 }) ∈ (Base‘𝐷)) → (((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 }) ↔ 𝑂 = (𝑉 × { 0 }))) |
19 | 14, 16, 18 | syl2anc 586 | . 2 ⊢ (𝜑 → (((𝑉 × { 0 })(+g‘𝐷)(𝑉 × { 0 })) = (𝑉 × { 0 }) ↔ 𝑂 = (𝑉 × { 0 }))) |
20 | 13, 19 | mpbid 234 | 1 ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {csn 4569 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 0gc0g 16715 Grpcgrp 18105 LModclmod 19636 LFnlclfn 36195 LDualcld 36261 |
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-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 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-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 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-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-sca 16583 df-vsca 16584 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-lfl 36196 df-ldual 36262 |
This theorem is referenced by: ldual0vcl 36289 lkr0f2 36299 lduallkr3 36300 lclkrlem1 38644 lclkrlem2j 38654 lcd0v 38749 lcd0v2 38750 |
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