Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lmodvsneg | 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 |
---|---|
lmodvsneg.b | ⊢ 𝐵 = (Base‘𝑊) |
lmodvsneg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsneg.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsneg.n | ⊢ 𝑁 = (invg‘𝑊) |
lmodvsneg.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodvsneg.m | ⊢ 𝑀 = (invg‘𝐹) |
lmodvsneg.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodvsneg.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
lmodvsneg.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
Ref | Expression |
---|---|
lmodvsneg | ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsneg.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodvsneg.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 19636 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring) |
5 | ringgrp 19296 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
7 | lmodvsneg.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
8 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 7, 8 | ringidcl 19312 | . . . . 5 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
10 | 4, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
11 | lmodvsneg.m | . . . . 5 ⊢ 𝑀 = (invg‘𝐹) | |
12 | 7, 11 | grpinvcl 18145 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
13 | 6, 10, 12 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
14 | lmodvsneg.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
15 | lmodvsneg.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | lmodvsneg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
17 | lmodvsneg.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
18 | eqid 2821 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 19653 | . . 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 1368 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
21 | 7, 18, 8, 11, 4, 14 | ringnegl 19338 | . . 3 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
22 | 21 | oveq1d 7165 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
23 | 16, 2, 17, 7 | lmodvscl 19645 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
24 | 1, 14, 15, 23 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝐵) |
25 | lmodvsneg.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 19671 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
27 | 1, 24, 26 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
28 | 20, 22, 27 | 3eqtr3rd 2865 | 1 ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 Grpcgrp 18097 invgcminusg 18098 1rcur 19245 Ringcrg 19291 LModclmod 19628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 |
This theorem is referenced by: lmodnegadd 19677 clmvsneg 23698 linds2eq 30936 baerlem5alem1 38838 lincext3 44504 lindslinindimp2lem4 44509 lincresunit3 44529 |
Copyright terms: Public domain | W3C validator |