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 2737 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | asclval 20839 | . . . 4 ⊢ (𝑅 ∈ 𝐾 → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
7 | 6 | 3ad2ant2 1136 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
8 | 7 | oveq1d 7228 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = ((𝑅 · (1r‘𝑊)) × 𝑋)) |
9 | simp1 1138 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ AssAlg) | |
10 | simp2 1139 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑅 ∈ 𝐾) | |
11 | assaring 20823 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
12 | 11 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Ring) |
13 | asclmul1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 5 | ringidcl 19586 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝑉) |
15 | 12, 14 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (1r‘𝑊) ∈ 𝑉) |
16 | simp3 1140 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
17 | asclmul1.t | . . . 4 ⊢ × = (.r‘𝑊) | |
18 | 13, 2, 3, 4, 17 | assaass 20820 | . . 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 1374 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝑅 · (1r‘𝑊)) × 𝑋) = (𝑅 · ((1r‘𝑊) × 𝑋))) |
20 | 13, 17, 5 | ringlidm 19589 | . . . 4 ⊢ ((𝑊 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
21 | 12, 16, 20 | syl2anc 587 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
22 | 21 | oveq2d 7229 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · ((1r‘𝑊) × 𝑋)) = (𝑅 · 𝑋)) |
23 | 8, 19, 22 | 3eqtrd 2781 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 .rcmulr 16803 Scalarcsca 16805 ·𝑠 cvsca 16806 1rcur 19516 Ringcrg 19562 AssAlgcasa 20812 algSccascl 20814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mgp 19505 df-ur 19517 df-ring 19564 df-assa 20815 df-ascl 20817 |
This theorem is referenced by: ascldimul 20847 issubassa2 20852 mplind 21028 evl1vsd 21260 evl1scvarpw 21279 chpscmatgsumbin 21741 fta1blem 25066 mhphf 39995 |
Copyright terms: Public domain | W3C validator |