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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mendvsca | Structured version Visualization version GIF version |
Description: A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
Ref | Expression |
---|---|
mendvscafval.a | ⊢ 𝐴 = (MEndo‘𝑀) |
mendvscafval.v | ⊢ · = ( ·𝑠 ‘𝑀) |
mendvscafval.b | ⊢ 𝐵 = (Base‘𝐴) |
mendvscafval.s | ⊢ 𝑆 = (Scalar‘𝑀) |
mendvscafval.k | ⊢ 𝐾 = (Base‘𝑆) |
mendvscafval.e | ⊢ 𝐸 = (Base‘𝑀) |
mendvsca.w | ⊢ ∙ = ( ·𝑠 ‘𝐴) |
Ref | Expression |
---|---|
mendvsca | ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘𝑓 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4331 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
2 | 1 | xpeq2d 5296 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐸 × {𝑥}) = (𝐸 × {𝑋})) |
3 | id 22 | . . 3 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
4 | 2, 3 | oveqan12d 6832 | . 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐸 × {𝑥}) ∘𝑓 · 𝑦) = ((𝐸 × {𝑋}) ∘𝑓 · 𝑌)) |
5 | mendvsca.w | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝐴) | |
6 | mendvscafval.a | . . . 4 ⊢ 𝐴 = (MEndo‘𝑀) | |
7 | mendvscafval.v | . . . 4 ⊢ · = ( ·𝑠 ‘𝑀) | |
8 | mendvscafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
9 | mendvscafval.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑀) | |
10 | mendvscafval.k | . . . 4 ⊢ 𝐾 = (Base‘𝑆) | |
11 | mendvscafval.e | . . . 4 ⊢ 𝐸 = (Base‘𝑀) | |
12 | 6, 7, 8, 9, 10, 11 | mendvscafval 38262 | . . 3 ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘𝑓 · 𝑦)) |
13 | 5, 12 | eqtri 2782 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘𝑓 · 𝑦)) |
14 | ovex 6841 | . 2 ⊢ ((𝐸 × {𝑋}) ∘𝑓 · 𝑌) ∈ V | |
15 | 4, 13, 14 | ovmpt2a 6956 | 1 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘𝑓 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 {csn 4321 × cxp 5264 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 ∘𝑓 cof 7060 Basecbs 16059 Scalarcsca 16146 ·𝑠 cvsca 16147 MEndocmend 38247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-lmhm 19224 df-mend 38248 |
This theorem is referenced by: mendlmod 38265 mendassa 38266 |
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