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Mirrors > Home > MPE Home > Th. List > asclmul1 | Structured version Visualization version GIF version |
Description: Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
asclmul1.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclmul1.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclmul1.k | ⊢ 𝐾 = (Base‘𝐹) |
asclmul1.v | ⊢ 𝑉 = (Base‘𝑊) |
asclmul1.t | ⊢ × = (.r‘𝑊) |
asclmul1.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
asclmul1 | ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asclmul1.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑊) | |
2 | asclmul1.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | asclmul1.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
4 | asclmul1.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | eqid 2726 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | asclval 21871 | . . . 4 ⊢ (𝑅 ∈ 𝐾 → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
7 | 6 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
8 | 7 | oveq1d 7429 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = ((𝑅 · (1r‘𝑊)) × 𝑋)) |
9 | simp1 1133 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ AssAlg) | |
10 | simp2 1134 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑅 ∈ 𝐾) | |
11 | assaring 21853 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
12 | 11 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Ring) |
13 | asclmul1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 5 | ringidcl 20239 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝑉) |
15 | 12, 14 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (1r‘𝑊) ∈ 𝑉) |
16 | simp3 1135 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
17 | asclmul1.t | . . . 4 ⊢ × = (.r‘𝑊) | |
18 | 13, 2, 3, 4, 17 | assaass 21850 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑅 ∈ 𝐾 ∧ (1r‘𝑊) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · (1r‘𝑊)) × 𝑋) = (𝑅 · ((1r‘𝑊) × 𝑋))) |
19 | 9, 10, 15, 16, 18 | syl13anc 1369 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝑅 · (1r‘𝑊)) × 𝑋) = (𝑅 · ((1r‘𝑊) × 𝑋))) |
20 | 13, 17, 5 | ringlidm 20242 | . . . 4 ⊢ ((𝑊 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
21 | 12, 16, 20 | syl2anc 582 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
22 | 21 | oveq2d 7430 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · ((1r‘𝑊) × 𝑋)) = (𝑅 · 𝑋)) |
23 | 8, 19, 22 | 3eqtrd 2770 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 .rcmulr 17260 Scalarcsca 17262 ·𝑠 cvsca 17263 1rcur 20158 Ringcrg 20210 AssAlgcasa 21842 algSccascl 21844 |
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 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-0g 17449 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mgp 20112 df-ur 20159 df-ring 20212 df-assa 21845 df-ascl 21847 |
This theorem is referenced by: ascldimul 21879 issubassa2 21883 mplind 22077 evl1vsd 22330 evl1scvarpw 22349 evls1fpws 22355 chpscmatgsumbin 22832 fta1blem 26193 aks5lem2 41897 selvvvval 42273 |
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