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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 19645 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring) |
5 | ringgrp 19305 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
7 | lmodvsneg.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
8 | eqid 2824 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 7, 8 | ringidcl 19321 | . . . . 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 18154 | . . . 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 2824 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
19 | 16, 2, 17, 7, 18 | lmodvsass 19662 | . . 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 19347 | . . 3 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
22 | 21 | oveq1d 7174 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
23 | 16, 2, 17, 7 | lmodvscl 19654 | . . . 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 19680 | . . 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 2868 | 1 ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 Scalarcsca 16571 ·𝑠 cvsca 16572 Grpcgrp 18106 invgcminusg 18107 1rcur 19254 Ringcrg 19300 LModclmod 19637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 |
This theorem is referenced by: lmodnegadd 19686 clmvsneg 23707 linds2eq 30945 baerlem5alem1 38848 lincext3 44518 lindslinindimp2lem4 44523 lincresunit3 44543 |
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