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Mirrors > Home > MPE Home > Th. List > sgsummulcl | Structured version Visualization version GIF version |
Description: A finite semiring sum multiplied by a constant, analogous to gsummulc2 19856. (Contributed by AV, 23-Aug-2019.) |
Ref | Expression |
---|---|
srgsummulcr.b | ⊢ 𝐵 = (Base‘𝑅) |
srgsummulcr.z | ⊢ 0 = (0g‘𝑅) |
srgsummulcr.p | ⊢ + = (+g‘𝑅) |
srgsummulcr.t | ⊢ · = (.r‘𝑅) |
srgsummulcr.r | ⊢ (𝜑 → 𝑅 ∈ SRing) |
srgsummulcr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
srgsummulcr.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
srgsummulcr.x | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
srgsummulcr.n | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) |
Ref | Expression |
---|---|
sgsummulcl | ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgsummulcr.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | srgsummulcr.z | . 2 ⊢ 0 = (0g‘𝑅) | |
3 | srgsummulcr.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
4 | srgcmn 19754 | . . 3 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
6 | srgmnd 19755 | . . 3 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
8 | srgsummulcr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | srgsummulcr.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | srgsummulcr.t | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | 1, 10 | srglmhm 19781 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 MndHom 𝑅)) |
12 | 3, 9, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑌 · 𝑥)) ∈ (𝑅 MndHom 𝑅)) |
13 | srgsummulcr.x | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
14 | srgsummulcr.n | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
15 | oveq2 7275 | . 2 ⊢ (𝑥 = 𝑋 → (𝑌 · 𝑥) = (𝑌 · 𝑋)) | |
16 | oveq2 7275 | . 2 ⊢ (𝑥 = (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝑌 · 𝑥) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | |
17 | 1, 2, 5, 7, 8, 12, 13, 14, 15, 16 | gsummhm2 19550 | 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 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5073 ↦ cmpt 5156 ‘cfv 6426 (class class class)co 7267 finSupp cfsupp 9115 Basecbs 16922 +gcplusg 16972 .rcmulr 16973 0gc0g 17160 Σg cgsu 17161 Mndcmnd 18395 MndHom cmhm 18438 CMndccmn 19396 SRingcsrg 19751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-fzo 13393 df-seq 13732 df-hash 14055 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-plusg 16985 df-0g 17162 df-gsum 17163 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-cntz 18933 df-cmn 19398 df-mgp 19731 df-srg 19752 |
This theorem is referenced by: (None) |
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