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Mirrors > Home > MPE Home > Th. List > lmodvsinv2 | Structured version Visualization version GIF version |
Description: Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
lmodvsinv2.b | ⊢ 𝐵 = (Base‘𝑊) |
lmodvsinv2.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsinv2.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsinv2.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodvsinv2.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
lmodvsinv2 | ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑁‘𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1136 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ LMod) | |
2 | lmodgrp 20329 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ Grp) |
4 | simp3 1138 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | lmodvsinv2.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
6 | eqid 2736 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | eqid 2736 | . . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | lmodvsinv2.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑊) | |
9 | 5, 6, 7, 8 | grprinv 18801 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑊)(𝑁‘𝑋)) = (0g‘𝑊)) |
10 | 3, 4, 9 | syl2anc 584 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑊)(𝑁‘𝑋)) = (0g‘𝑊)) |
11 | 10 | oveq2d 7373 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑋(+g‘𝑊)(𝑁‘𝑋))) = (𝑅 · (0g‘𝑊))) |
12 | simp2 1137 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ 𝐾) | |
13 | 5, 8 | grpinvcl 18798 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
14 | 3, 4, 13 | syl2anc 584 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
15 | lmodvsinv2.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
16 | lmodvsinv2.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | lmodvsinv2.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
18 | 5, 6, 15, 16, 17 | lmodvsdi 20345 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵)) → (𝑅 · (𝑋(+g‘𝑊)(𝑁‘𝑋))) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · (𝑁‘𝑋)))) |
19 | 1, 12, 4, 14, 18 | syl13anc 1372 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑋(+g‘𝑊)(𝑁‘𝑋))) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · (𝑁‘𝑋)))) |
20 | 15, 16, 17, 7 | lmodvs0 20356 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑅 · (0g‘𝑊)) = (0g‘𝑊)) |
21 | 1, 12, 20 | syl2anc 584 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (0g‘𝑊)) = (0g‘𝑊)) |
22 | 11, 19, 21 | 3eqtr3d 2784 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · (𝑁‘𝑋))) = (0g‘𝑊)) |
23 | 5, 15, 16, 17 | lmodvscl 20339 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
24 | 5, 15, 16, 17 | lmodvscl 20339 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑅 · (𝑁‘𝑋)) ∈ 𝐵) |
25 | 1, 12, 14, 24 | syl3anc 1371 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑁‘𝑋)) ∈ 𝐵) |
26 | 5, 6, 7, 8 | grpinvid1 18802 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ (𝑅 · 𝑋) ∈ 𝐵 ∧ (𝑅 · (𝑁‘𝑋)) ∈ 𝐵) → ((𝑁‘(𝑅 · 𝑋)) = (𝑅 · (𝑁‘𝑋)) ↔ ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · (𝑁‘𝑋))) = (0g‘𝑊))) |
27 | 3, 23, 25, 26 | syl3anc 1371 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑁‘(𝑅 · 𝑋)) = (𝑅 · (𝑁‘𝑋)) ↔ ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · (𝑁‘𝑋))) = (0g‘𝑊))) |
28 | 22, 27 | mpbird 256 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑅 · 𝑋)) = (𝑅 · (𝑁‘𝑋))) |
29 | 28 | eqcomd 2742 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑁‘𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 Grpcgrp 18748 invgcminusg 18749 LModclmod 20322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-mgp 19897 df-ring 19966 df-lmod 20324 |
This theorem is referenced by: invlmhm 20503 eqgvscpbl 32142 |
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