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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 20751 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| 4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring) |
| 5 | ringgrp 20178 | . . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Grp) |
| 7 | lmodvsneg.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
| 8 | eqid 2728 | . . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 9 | 7, 8 | ringidcl 20202 | . . . . 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 18944 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ (1r‘𝐹) ∈ 𝐾) → (𝑀‘(1r‘𝐹)) ∈ 𝐾) |
| 13 | 6, 10, 12 | syl2anc 583 | . . 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 2728 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 19 | 16, 2, 17, 7, 18 | lmodvsass 20770 | . . 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 1370 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋))) |
| 21 | 7, 18, 8, 11, 4, 14 | ringnegl 20238 | . . 3 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) = (𝑀‘𝑅)) |
| 22 | 21 | oveq1d 7435 | . 2 ⊢ (𝜑 → (((𝑀‘(1r‘𝐹))(.r‘𝐹)𝑅) · 𝑋) = ((𝑀‘𝑅) · 𝑋)) |
| 23 | 16, 2, 17, 7 | lmodvscl 20761 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · 𝑋) ∈ 𝐵) |
| 24 | 1, 14, 15, 23 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝐵) |
| 25 | lmodvsneg.n | . . . 4 ⊢ 𝑁 = (invg‘𝑊) | |
| 26 | 16, 25, 2, 17, 8, 11 | lmodvneg1 20788 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝐵) → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
| 27 | 1, 24, 26 | syl2anc 583 | . 2 ⊢ (𝜑 → ((𝑀‘(1r‘𝐹)) · (𝑅 · 𝑋)) = (𝑁‘(𝑅 · 𝑋))) |
| 28 | 20, 22, 27 | 3eqtr3rd 2777 | 1 ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 .rcmulr 17234 Scalarcsca 17236 ·𝑠 cvsca 17237 Grpcgrp 18890 invgcminusg 18891 1rcur 20121 Ringcrg 20173 LModclmod 20743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-lmod 20745 |
| This theorem is referenced by: lmodnegadd 20794 clmvsneg 25040 linds2eq 33109 baerlem5alem1 41181 lincext3 47524 lindslinindimp2lem4 47529 lincresunit3 47549 |
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