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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 2824 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | asclval 20112 | . . . 4 ⊢ (𝑅 ∈ 𝐾 → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
7 | 6 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐴‘𝑅) = (𝑅 · (1r‘𝑊))) |
8 | 7 | oveq1d 7174 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = ((𝑅 · (1r‘𝑊)) × 𝑋)) |
9 | simp1 1132 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ AssAlg) | |
10 | simp2 1133 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑅 ∈ 𝐾) | |
11 | assaring 20096 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
12 | 11 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑊 ∈ Ring) |
13 | asclmul1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 5 | ringidcl 19321 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ 𝑉) |
15 | 12, 14 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (1r‘𝑊) ∈ 𝑉) |
16 | simp3 1134 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
17 | asclmul1.t | . . . 4 ⊢ × = (.r‘𝑊) | |
18 | 13, 2, 3, 4, 17 | assaass 20093 | . . 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 1368 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝑅 · (1r‘𝑊)) × 𝑋) = (𝑅 · ((1r‘𝑊) × 𝑋))) |
20 | 13, 17, 5 | ringlidm 19324 | . . . 4 ⊢ ((𝑊 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
21 | 12, 16, 20 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((1r‘𝑊) × 𝑋) = 𝑋) |
22 | 21 | oveq2d 7175 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · ((1r‘𝑊) × 𝑋)) = (𝑅 · 𝑋)) |
23 | 8, 19, 22 | 3eqtrd 2863 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → ((𝐴‘𝑅) × 𝑋) = (𝑅 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 Scalarcsca 16571 ·𝑠 cvsca 16572 1rcur 19254 Ringcrg 19300 AssAlgcasa 20085 algSccascl 20087 |
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-rep 5193 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-mgp 19243 df-ur 19255 df-ring 19302 df-assa 20088 df-ascl 20090 |
This theorem is referenced by: ascldimul 20119 issubassa2 20124 mplind 20285 evl1vsd 20510 evl1scvarpw 20529 chpscmatgsumbin 21455 fta1blem 24765 |
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