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Mirrors > Home > MPE Home > Th. List > psrvscaval | 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) |
psrvscaval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
psrvscaval | ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrvsca.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | psrvsca.n | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑆) | |
3 | psrvsca.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | psrvsca.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
5 | psrvsca.m | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | psrvsca.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | psrvsca.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | psrvsca.y | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | psrvsca 20770 | . . 3 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
10 | 9 | fveq1d 6676 | . 2 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌)) |
11 | psrvscaval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | ovex 7203 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 6, 12 | rabex2 5202 | . . . . 5 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | 1, 3, 6, 4, 8 | psrelbas 20758 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶𝐾) |
16 | 15 | ffnd 6505 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
17 | eqidd 2739 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) = (𝐹‘𝑌)) | |
18 | 14, 7, 16, 17 | ofc1 7450 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
19 | 11, 18 | mpdan 687 | . 2 ⊢ (𝜑 → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
20 | 10, 19 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3057 Vcvv 3398 {csn 4516 × cxp 5523 ◡ccnv 5524 “ cima 5528 ‘cfv 6339 (class class class)co 7170 ∘f cof 7423 ↑m cmap 8437 Fincfn 8555 ℕcn 11716 ℕ0cn0 11976 Basecbs 16586 .rcmulr 16669 ·𝑠 cvsca 16672 mPwSer cmps 20717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-tset 16687 df-psr 20722 |
This theorem is referenced by: psrass23l 20787 psrass23 20789 mpllsslem 20816 |
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