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Mirrors > Home > MPE Home > Th. List > gsumvsmul | Structured version Visualization version GIF version |
Description: Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 18961, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) (Revised by AV, 10-Jul-2019.) |
Ref | Expression |
---|---|
gsumvsmul.b | ⊢ 𝐵 = (Base‘𝑅) |
gsumvsmul.s | ⊢ 𝑆 = (Scalar‘𝑅) |
gsumvsmul.k | ⊢ 𝐾 = (Base‘𝑆) |
gsumvsmul.z | ⊢ 0 = (0g‘𝑅) |
gsumvsmul.p | ⊢ + = (+g‘𝑅) |
gsumvsmul.t | ⊢ · = ( ·𝑠 ‘𝑅) |
gsumvsmul.r | ⊢ (𝜑 → 𝑅 ∈ LMod) |
gsumvsmul.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumvsmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
gsumvsmul.y | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
gsumvsmul.n | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) |
Ref | Expression |
---|---|
gsumvsmul | ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = (𝑋 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumvsmul.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | gsumvsmul.z | . 2 ⊢ 0 = (0g‘𝑅) | |
3 | gsumvsmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ LMod) | |
4 | lmodcmn 19267 | . . 3 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ CMnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
6 | cmnmnd 18561 | . . 3 ⊢ (𝑅 ∈ CMnd → 𝑅 ∈ Mnd) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
8 | gsumvsmul.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsumvsmul.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
10 | gsumvsmul.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑅) | |
11 | gsumvsmul.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑅) | |
12 | gsumvsmul.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
13 | 1, 10, 11, 12 | lmodvsghm 19280 | . . . 4 ⊢ ((𝑅 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑦 ∈ 𝐵 ↦ (𝑋 · 𝑦)) ∈ (𝑅 GrpHom 𝑅)) |
14 | 3, 9, 13 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋 · 𝑦)) ∈ (𝑅 GrpHom 𝑅)) |
15 | ghmmhm 18021 | . . 3 ⊢ ((𝑦 ∈ 𝐵 ↦ (𝑋 · 𝑦)) ∈ (𝑅 GrpHom 𝑅) → (𝑦 ∈ 𝐵 ↦ (𝑋 · 𝑦)) ∈ (𝑅 MndHom 𝑅)) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋 · 𝑦)) ∈ (𝑅 MndHom 𝑅)) |
17 | gsumvsmul.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
18 | gsumvsmul.n | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) | |
19 | oveq2 6913 | . 2 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
20 | oveq2 6913 | . 2 ⊢ (𝑦 = (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)) → (𝑋 · 𝑦) = (𝑋 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)))) | |
21 | 1, 2, 5, 7, 8, 16, 17, 18, 19, 20 | gsummhm2 18692 | 1 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = (𝑋 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 finSupp cfsupp 8544 Basecbs 16222 +gcplusg 16305 Scalarcsca 16308 ·𝑠 cvsca 16309 0gc0g 16453 Σg cgsu 16454 Mndcmnd 17647 MndHom cmhm 17686 GrpHom cghm 18008 CMndccmn 18546 LModclmod 19219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-ghm 18009 df-cntz 18100 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-lmod 19221 |
This theorem is referenced by: frlmup1 20504 lincscm 43066 lincresunit3lem2 43116 |
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