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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 21963 | . . 3 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘f − 𝑘))))))) | 
| 9 | 8 | fveq1d 6908 | . 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 5147 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∘r ≤ 𝑥 ↔ 𝑦 ∘r ≤ 𝑋)) | |
| 12 | 11 | rabbidv 3444 | . . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋}) | 
| 13 | fvoveq1 7454 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝑥 ∘f − 𝑘)) = (𝐺‘(𝑋 ∘f − 𝑘))) | |
| 14 | 13 | oveq2d 7447 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘f − 𝑘))) = ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘f − 𝑘)))) | 
| 15 | 12, 14 | mpteq12dv 5233 | . . . . 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))))) | 
| 16 | 15 | oveq2d 7447 | . . . 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)))))) | 
| 17 | eqid 2737 | . . . 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)))))) | |
| 18 | ovex 7464 | . . . 4 ⊢ (𝑅 Σg (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘f − 𝑘))))) ∈ V | |
| 19 | 16, 17, 18 | fvmpt 7016 | . . 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)))))) | 
| 20 | 10, 19 | 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)))))) | 
| 21 | 9, 20 | eqtrd 2777 | 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 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ∘r cofr 7696 ↑m cmap 8866 Fincfn 8985 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 Σg cgsu 17485 mPwSer cmps 21924 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-psr 21929 | 
| This theorem is referenced by: psrlidm 21982 psrridm 21983 psrass1 21984 mplsubrglem 22024 psdmul 22170 | 
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