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Mirrors > Home > MPE Home > Th. List > psrvsca | Structured version Visualization version GIF version |
Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
psrvsca.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrvsca.n | ⊢ ∙ = ( ·𝑠 ‘𝑆) |
psrvsca.k | ⊢ 𝐾 = (Base‘𝑅) |
psrvsca.b | ⊢ 𝐵 = (Base‘𝑆) |
psrvsca.m | ⊢ · = (.r‘𝑅) |
psrvsca.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrvsca.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
psrvsca.y | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
psrvsca | ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrvsca.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
2 | psrvsca.y | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
3 | sneq 4331 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
4 | 3 | xpeq2d 5296 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐷 × {𝑥}) = (𝐷 × {𝑋})) |
5 | 4 | oveq1d 6829 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = ((𝐷 × {𝑋}) ∘𝑓 · 𝑓)) |
6 | oveq2 6822 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝐷 × {𝑋}) ∘𝑓 · 𝑓) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹)) | |
7 | psrvsca.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | psrvsca.n | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑆) | |
9 | psrvsca.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
10 | psrvsca.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
11 | psrvsca.m | . . . 4 ⊢ · = (.r‘𝑅) | |
12 | psrvsca.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
13 | 7, 8, 9, 10, 11, 12 | psrvscafval 19612 | . . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) |
14 | ovex 6842 | . . 3 ⊢ ((𝐷 × {𝑋}) ∘𝑓 · 𝐹) ∈ V | |
15 | 5, 6, 13, 14 | ovmpt2 6962 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝐹 ∈ 𝐵) → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹)) |
16 | 1, 2, 15 | syl2anc 696 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 {crab 3054 {csn 4321 × cxp 5264 ◡ccnv 5265 “ cima 5269 ‘cfv 6049 (class class class)co 6814 ∘𝑓 cof 7061 ↑𝑚 cmap 8025 Fincfn 8123 ℕcn 11232 ℕ0cn0 11504 Basecbs 16079 .rcmulr 16164 ·𝑠 cvsca 16167 mPwSer cmps 19573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-tset 16182 df-psr 19578 |
This theorem is referenced by: psrvscaval 19614 psrvscacl 19615 psrlmod 19623 psrass23l 19630 psrass23 19632 resspsrvsca 19640 mplvsca 19669 |
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