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Mirrors > Home > MPE Home > Th. List > psrmulval | Structured version Visualization version GIF version |
Description: The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
psrmulr.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrmulr.b | ⊢ 𝐵 = (Base‘𝑆) |
psrmulr.m | ⊢ · = (.r‘𝑅) |
psrmulr.t | ⊢ ∙ = (.r‘𝑆) |
psrmulr.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrmulfval.i | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
psrmulfval.r | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
psrmulval.r | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
psrmulval | ⊢ (𝜑 → ((𝐹 ∙ 𝐺)‘𝑋) = (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrmulr.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | psrmulr.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
3 | psrmulr.m | . . . 4 ⊢ · = (.r‘𝑅) | |
4 | psrmulr.t | . . . 4 ⊢ ∙ = (.r‘𝑆) | |
5 | psrmulr.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | psrmulfval.i | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | psrmulfval.r | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | psrmulfval 19385 | . . 3 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))))) |
9 | 8 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((𝐹 ∙ 𝐺)‘𝑋) = ((𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))))‘𝑋)) |
10 | psrmulval.r | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
11 | breq2 4657 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∘𝑟 ≤ 𝑥 ↔ 𝑦 ∘𝑟 ≤ 𝑋)) | |
12 | 11 | rabbidv 3189 | . . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋}) |
13 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑓 − 𝑘) = (𝑋 ∘𝑓 − 𝑘)) | |
14 | 13 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝑥 ∘𝑓 − 𝑘)) = (𝐺‘(𝑋 ∘𝑓 − 𝑘))) |
15 | 14 | oveq2d 6666 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))) = ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))) |
16 | 12, 15 | mpteq12dv 4733 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘)))) = (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘))))) |
17 | 16 | oveq2d 6666 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))))) |
18 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘)))))) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘)))))) | |
19 | ovex 6678 | . . . 4 ⊢ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘))))) ∈ V | |
20 | 17, 18, 19 | fvmpt 6282 | . . 3 ⊢ (𝑋 ∈ 𝐷 → ((𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))))‘𝑋) = (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))))) |
21 | 10, 20 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))))‘𝑋) = (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))))) |
22 | 9, 21 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝐹 ∙ 𝐺)‘𝑋) = (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 class class class wbr 4653 ↦ cmpt 4729 ◡ccnv 5113 “ cima 5117 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ∘𝑟 cofr 6896 ↑𝑚 cmap 7857 Fincfn 7955 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 .rcmulr 15942 Σg cgsu 16101 mPwSer cmps 19351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-psr 19356 |
This theorem is referenced by: psrlidm 19403 psrridm 19404 psrass1 19405 mplsubrglem 19439 |
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