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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 4570 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
2 | 1 | xpeq2d 5579 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐸 × {𝑥}) = (𝐸 × {𝑋})) |
3 | id 22 | . . 3 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
4 | 2, 3 | oveqan12d 7169 | . 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 39783 | . . 3 ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) |
13 | 5, 12 | eqtri 2844 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) |
14 | ovex 7183 | . 2 ⊢ ((𝐸 × {𝑋}) ∘f · 𝑌) ∈ V | |
15 | 4, 13, 14 | ovmpoa 7299 | 1 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘f · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4560 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ∘f cof 7401 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 MEndocmend 39768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-lmhm 19788 df-mend 39769 |
This theorem is referenced by: mendlmod 39786 mendassa 39787 |
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