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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 19206 | . . 3 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))))))) |
9 | 8 | fveq1d 6105 | . 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 4587 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∘𝑟 ≤ 𝑥 ↔ 𝑦 ∘𝑟 ≤ 𝑋)) | |
12 | 11 | rabbidv 3164 | . . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑥} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋}) |
13 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑓 − 𝑘) = (𝑋 ∘𝑓 − 𝑘)) | |
14 | 13 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝑥 ∘𝑓 − 𝑘)) = (𝐺‘(𝑋 ∘𝑓 − 𝑘))) |
15 | 14 | oveq2d 6565 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘𝑓 − 𝑘))) = ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘)))) |
16 | 12, 15 | mpteq12dv 4663 | . . . . 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 6565 | . . . 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 2610 | . . . 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 6577 | . . . 4 ⊢ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘𝑓 − 𝑘))))) ∈ V | |
20 | 17, 18, 19 | fvmpt 6191 | . . 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 2644 | 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 1475 ∈ wcel 1977 {crab 2900 class class class wbr 4583 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ∘𝑟 cofr 6794 ↑𝑚 cmap 7744 Fincfn 7841 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 .rcmulr 15769 Σg cgsu 15924 mPwSer cmps 19172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-psr 19177 |
This theorem is referenced by: psrlidm 19224 psrridm 19225 psrass1 19226 mplsubrglem 19260 |
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