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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 20165 | . . 3 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
10 | 9 | fveq1d 6667 | . 2 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌)) |
11 | psrvscaval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | ovex 7183 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 6, 12 | rabex2 5230 | . . . . 5 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | 1, 3, 6, 4, 8 | psrelbas 20153 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶𝐾) |
16 | 15 | ffnd 6510 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
17 | eqidd 2822 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) = (𝐹‘𝑌)) | |
18 | 14, 7, 16, 17 | ofc1 7426 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
19 | 11, 18 | mpdan 685 | . 2 ⊢ (𝜑 → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
20 | 10, 19 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 {csn 4561 × cxp 5548 ◡ccnv 5549 “ cima 5553 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 ↑m cmap 8400 Fincfn 8503 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 .rcmulr 16560 ·𝑠 cvsca 16563 mPwSer cmps 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-psr 20130 |
This theorem is referenced by: psrass23l 20182 psrass23 20184 mpllsslem 20209 |
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