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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 4567 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
4 | 3 | xpeq2d 5598 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐷 × {𝑥}) = (𝐷 × {𝑋})) |
5 | 4 | oveq1d 7249 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐷 × {𝑥}) ∘f · 𝑓) = ((𝐷 × {𝑋}) ∘f · 𝑓)) |
6 | oveq2 7242 | . . 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 20944 | . . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)) |
14 | ovex 7267 | . . 3 ⊢ ((𝐷 × {𝑋}) ∘f · 𝐹) ∈ V | |
15 | 5, 6, 13, 14 | ovmpo 7390 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝐹 ∈ 𝐵) → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
16 | 1, 2, 15 | syl2anc 587 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {crab 3067 {csn 4557 × cxp 5566 ◡ccnv 5567 “ cima 5571 ‘cfv 6400 (class class class)co 7234 ∘f cof 7488 ↑m cmap 8531 Fincfn 8649 ℕcn 11857 ℕ0cn0 12117 Basecbs 16790 .rcmulr 16833 ·𝑠 cvsca 16836 mPwSer cmps 20892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10812 ax-resscn 10813 ax-1cn 10814 ax-icn 10815 ax-addcl 10816 ax-addrcl 10817 ax-mulcl 10818 ax-mulrcl 10819 ax-mulcom 10820 ax-addass 10821 ax-mulass 10822 ax-distr 10823 ax-i2m1 10824 ax-1ne0 10825 ax-1rid 10826 ax-rnegex 10827 ax-rrecex 10828 ax-cnre 10829 ax-pre-lttri 10830 ax-pre-lttrn 10831 ax-pre-ltadd 10832 ax-pre-mulgt0 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-of 7490 df-om 7666 df-1st 7782 df-2nd 7783 df-supp 7927 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-er 8414 df-map 8533 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-fsupp 9013 df-pnf 10896 df-mnf 10897 df-xr 10898 df-ltxr 10899 df-le 10900 df-sub 11091 df-neg 11092 df-nn 11858 df-2 11920 df-3 11921 df-4 11922 df-5 11923 df-6 11924 df-7 11925 df-8 11926 df-9 11927 df-n0 12118 df-z 12204 df-uz 12466 df-fz 13123 df-struct 16730 df-slot 16765 df-ndx 16775 df-base 16791 df-plusg 16845 df-mulr 16846 df-sca 16848 df-vsca 16849 df-tset 16851 df-psr 20897 |
This theorem is referenced by: psrvscaval 20946 psrvscacl 20947 psrlmod 20955 psrass23l 20962 psrass23 20964 resspsrvsca 20972 mplvsca 21004 |
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