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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsub | Structured version Visualization version GIF version |
Description: The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
ldualvsub.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsub.n | ⊢ 𝑁 = (invg‘𝑅) |
ldualvsub.u | ⊢ 1 = (1r‘𝑅) |
ldualvsub.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsub.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsub.p | ⊢ + = (+g‘𝐷) |
ldualvsub.t | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsub.m | ⊢ − = (-g‘𝐷) |
ldualvsub.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsub.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvsub.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsub | ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsub.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
2 | ldualvsub.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | 1, 2 | lduallmod 33458 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
4 | ldualvsub.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | eqid 2610 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | ldualvsub.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | 4, 1, 5, 2, 6 | ldualelvbase 33432 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
8 | ldualvsub.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 4, 1, 5, 2, 8 | ldualelvbase 33432 | . . 3 ⊢ (𝜑 → 𝐻 ∈ (Base‘𝐷)) |
10 | ldualvsub.p | . . . 4 ⊢ + = (+g‘𝐷) | |
11 | ldualvsub.m | . . . 4 ⊢ − = (-g‘𝐷) | |
12 | eqid 2610 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
13 | ldualvsub.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
14 | eqid 2610 | . . . 4 ⊢ (invg‘(Scalar‘𝐷)) = (invg‘(Scalar‘𝐷)) | |
15 | eqid 2610 | . . . 4 ⊢ (1r‘(Scalar‘𝐷)) = (1r‘(Scalar‘𝐷)) | |
16 | 5, 10, 11, 12, 13, 14, 15 | lmodvsubval2 18741 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷) ∧ 𝐻 ∈ (Base‘𝐷)) → (𝐺 − 𝐻) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
17 | 3, 7, 9, 16 | syl3anc 1318 | . 2 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
18 | ldualvsub.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑊) | |
19 | eqid 2610 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
20 | 18, 19, 1, 12, 2 | ldualsca 33437 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝐷) = (oppr‘𝑅)) |
21 | 20 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (invg‘(Scalar‘𝐷)) = (invg‘(oppr‘𝑅))) |
22 | ldualvsub.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
23 | 19, 22 | opprneg 18458 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
24 | 21, 23 | syl6reqr 2663 | . . . . 5 ⊢ (𝜑 → 𝑁 = (invg‘(Scalar‘𝐷))) |
25 | 20 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝐷)) = (1r‘(oppr‘𝑅))) |
26 | ldualvsub.u | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
27 | 19, 26 | oppr1 18457 | . . . . . 6 ⊢ 1 = (1r‘(oppr‘𝑅)) |
28 | 25, 27 | syl6reqr 2663 | . . . . 5 ⊢ (𝜑 → 1 = (1r‘(Scalar‘𝐷))) |
29 | 24, 28 | fveq12d 6109 | . . . 4 ⊢ (𝜑 → (𝑁‘ 1 ) = ((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷)))) |
30 | 29 | oveq1d 6564 | . . 3 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝐻) = (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻)) |
31 | 30 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝐺 + ((𝑁‘ 1 ) · 𝐻)) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
32 | 17, 31 | eqtr4d 2647 | 1 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 invgcminusg 17246 -gcsg 17247 1rcur 18324 opprcoppr 18445 LModclmod 18686 LFnlclfn 33362 LDualcld 33428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-lmod 18688 df-lfl 33363 df-ldual 33429 |
This theorem is referenced by: ldualvsubcl 33461 lcfrlem2 35850 |
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