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Mirrors > Home > MPE Home > Th. List > gsummulc1 | 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.) Remove unused hypothesis. (Revised by SN, 7-Mar-2025.) |
Ref | Expression |
---|---|
gsummulc1.b | ⊢ 𝐵 = (Base‘𝑅) |
gsummulc1.z | ⊢ 0 = (0g‘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 |
---|---|
gsummulc1 | ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) |
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 | 3 | ringcmnd 20219 | . 2 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
5 | ringmnd 20182 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
7 | gsummulc1.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsummulc1.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | gsummulc1.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
10 | 1, 9 | ringrghm 20248 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑌)) ∈ (𝑅 GrpHom 𝑅)) |
11 | 3, 8, 10 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑌)) ∈ (𝑅 GrpHom 𝑅)) |
12 | ghmmhm 19179 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑌)) ∈ (𝑅 GrpHom 𝑅) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑌)) ∈ (𝑅 MndHom 𝑅)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑌)) ∈ (𝑅 MndHom 𝑅)) |
14 | gsummulc1.x | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
15 | gsummulc1.n | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
16 | oveq1 7427 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑌) = (𝑋 · 𝑌)) | |
17 | oveq1 7427 | . 2 ⊢ (𝑥 = (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝑥 · 𝑌) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) | |
18 | 1, 2, 4, 6, 7, 13, 14, 15, 16, 17 | gsummhm2 19893 | 1 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 finSupp cfsupp 9385 Basecbs 17179 .rcmulr 17233 0gc0g 17420 Σg cgsu 17421 Mndcmnd 18693 MndHom cmhm 18737 GrpHom cghm 19166 Ringcrg 20172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18892 df-minusg 18893 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-ur 20121 df-ring 20174 |
This theorem is referenced by: gsumdixp 20254 psrass1 21906 mamuass 22301 mavmulass 22450 fedgmullem1 33323 fedgmullem2 33324 evlselv 41820 |
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