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Mirrors > Home > MPE Home > Th. List > lmodvsinv | Structured version Visualization version GIF version |
Description: Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
lmodvsinv.b | ⊢ 𝐵 = (Base‘𝑊) |
lmodvsinv.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsinv.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsinv.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodvsinv.m | ⊢ 𝑀 = (invg‘𝐹) |
lmodvsinv.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
lmodvsinv | ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑅) · 𝑋) = (𝑁‘(𝑅 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1172 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ LMod) | |
2 | lmodvsinv.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 19227 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 3 | 3ad2ant1 1169 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝐹 ∈ Ring) |
5 | ringgrp 18906 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝐹 ∈ Grp) |
7 | lmodvsinv.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
8 | eqid 2825 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 7, 8 | ringidcl 18922 | . . . . 5 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
10 | 4, 9 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (1r‘𝐹) ∈ 𝐾) |
11 | lmodvsinv.m | . . . . 5 ⊢ 𝑀 = (invg‘𝐹) | |
12 | 7, 11 | grpinvcl 17821 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
13 | 6, 10, 12 | syl2anc 581 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
14 | simp2 1173 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ 𝐾) | |
15 | simp3 1174 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
16 | lmodvsinv.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
17 | lmodvsinv.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
18 | eqid 2825 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 19244 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ ((𝑀‘(1r‘𝐹)) ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
20 | 1, 13, 14, 15, 19 | syl13anc 1497 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 18948 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
22 | 21 | oveq1d 6920 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
23 | 16, 2, 17, 7 | lmodvscl 19236 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
24 | lmodvsinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
25 | 16, 24, 2, 17, 8, 11 | lmodvneg1 19262 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
26 | 1, 23, 25 | syl2anc 581 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
27 | 20, 22, 26 | 3eqtr3d 2869 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑅) · 𝑋) = (𝑁‘(𝑅 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 .rcmulr 16306 Scalarcsca 16308 ·𝑠 cvsca 16309 Grpcgrp 17776 invgcminusg 17777 1rcur 18855 Ringcrg 18901 LModclmod 19219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-mgp 18844 df-ur 18856 df-ring 18903 df-lmod 19221 |
This theorem is referenced by: islindf4 20544 |
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