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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 19327 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
6 | ringmnd 19300 | . . 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 19350 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
12 | 3, 9, 11 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) |
13 | ghmmhm 18360 | . . 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 7143 | . 2 ⊢ (𝑥 = 𝑋 → (𝑌 · 𝑥) = (𝑌 · 𝑋)) | |
18 | oveq2 7143 | . 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 19052 | 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 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Σg cgsu 16706 Mndcmnd 17903 MndHom cmhm 17946 GrpHom cghm 18347 CMndccmn 18898 Ringcrg 19290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 |
This theorem is referenced by: gsumdixp 19355 frlmphl 20470 psrass1 20643 psrass23l 20646 psrass23 20648 mamuass 21007 mamuvs1 21010 mamuvs2 21011 mavmulass 21154 mdetrsca 21208 cpmadugsumlemB 21479 cpmadugsumlemC 21480 amgmlem 25575 mdetpmtr1 31176 matunitlindflem1 35053 |
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