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Mirrors > Home > MPE Home > Th. List > gsummulc2 | Structured version Visualization version GIF version |
Description: A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) |
Ref | Expression |
---|---|
gsummulc1.b | ⊢ 𝐵 = (Base‘𝑅) |
gsummulc1.z | ⊢ 0 = (0g‘𝑅) |
gsummulc1.p | ⊢ + = (+g‘𝑅) |
gsummulc1.t | ⊢ · = (.r‘𝑅) |
gsummulc1.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
gsummulc1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummulc1.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsummulc1.x | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsummulc1.n | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) |
Ref | Expression |
---|---|
gsummulc2 | ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummulc1.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | gsummulc1.z | . 2 ⊢ 0 = (0g‘𝑅) | |
3 | gsummulc1.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | ringcmn 18942 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
6 | ringmnd 18917 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
8 | gsummulc1.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsummulc1.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | gsummulc1.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
11 | 1, 10 | ringlghm 18965 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
12 | 3, 9, 11 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
13 | ghmmhm 18028 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 GrpHom 𝑅) → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 MndHom 𝑅)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 MndHom 𝑅)) |
15 | gsummulc1.x | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
16 | gsummulc1.n | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
17 | oveq2 6918 | . 2 ⊢ (𝑥 = 𝑋 → (𝑌 · 𝑥) = (𝑌 · 𝑋)) | |
18 | oveq2 6918 | . 2 ⊢ (𝑥 = (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝑌 · 𝑥) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | |
19 | 1, 2, 5, 7, 8, 14, 15, 16, 17, 18 | gsummhm2 18699 | 1 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ↦ cmpt 4954 ‘cfv 6127 (class class class)co 6910 finSupp cfsupp 8550 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 0gc0g 16460 Σg cgsu 16461 Mndcmnd 17654 MndHom cmhm 17693 GrpHom cghm 18015 CMndccmn 18553 Ringcrg 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 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 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-0g 16462 df-gsum 16463 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 |
This theorem is referenced by: gsumdixp 18970 psrass1 19773 psrass23l 19776 psrass23 19778 frlmphl 20494 mamuass 20582 mamuvs1 20585 mamuvs2 20586 mavmulass 20730 mdetrsca 20784 cpmadugsumlemB 21056 cpmadugsumlemC 21057 amgmlem 25136 mdetpmtr1 30430 matunitlindflem1 33944 |
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