![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2780 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | asclval 19841 | . . . 4 ⊢ (𝑅 ∈ 𝐾 → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
7 | 6 | 3ad2ant2 1115 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
8 | 7 | oveq1d 6997 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = ((𝑅 · (1r‘𝑊)) × 𝑋)) |
9 | simp1 1117 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ AssAlg) | |
10 | simp2 1118 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑅 ∈ 𝐾) | |
11 | assaring 19826 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
12 | 11 | 3ad2ant1 1114 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Ring) |
13 | asclmul1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 5 | ringidcl 19053 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝑉) |
15 | 12, 14 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (1r‘𝑊) ∈ 𝑉) |
16 | simp3 1119 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
17 | asclmul1.t | . . . 4 ⊢ × = (.r‘𝑊) | |
18 | 13, 2, 3, 4, 17 | assaass 19823 | . . 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 1353 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝑅 · (1r‘𝑊)) × 𝑋) = (𝑅 · ((1r‘𝑊) × 𝑋))) |
20 | 13, 17, 5 | ringlidm 19056 | . . . 4 ⊢ ((𝑊 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
21 | 12, 16, 20 | syl2anc 576 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
22 | 21 | oveq2d 6998 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · ((1r‘𝑊) × 𝑋)) = (𝑅 · 𝑋)) |
23 | 8, 19, 22 | 3eqtrd 2820 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ‘cfv 6193 (class class class)co 6982 Basecbs 16345 .rcmulr 16428 Scalarcsca 16430 ·𝑠 cvsca 16431 1rcur 18986 Ringcrg 19032 AssAlgcasa 19815 algSccascl 19817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-plusg 16440 df-0g 16577 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mgp 18975 df-ur 18987 df-ring 19034 df-assa 19818 df-ascl 19820 |
This theorem is referenced by: issubassa2 19851 mplind 20007 evl1vsd 20224 evl1scvarpw 20243 chpscmatgsumbin 21171 fta1blem 24480 |
Copyright terms: Public domain | W3C validator |