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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 4634 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
4 | 3 | xpeq2d 5702 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐷 × {𝑥}) = (𝐷 × {𝑋})) |
5 | 4 | oveq1d 7429 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐷 × {𝑥}) ∘f · 𝑓) = ((𝐷 × {𝑋}) ∘f · 𝑓)) |
6 | oveq2 7422 | . . 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 21884 | . . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)) |
14 | ovex 7447 | . . 3 ⊢ ((𝐷 × {𝑋}) ∘f · 𝐹) ∈ V | |
15 | 5, 6, 13, 14 | ovmpo 7575 | . 2 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝐹 ∈ 𝐵) → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
16 | 1, 2, 15 | syl2anc 583 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3427 {csn 4624 × cxp 5670 ◡ccnv 5671 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ∘f cof 7677 ↑m cmap 8838 Fincfn 8957 ℕcn 12236 ℕ0cn0 12496 Basecbs 17173 .rcmulr 17227 ·𝑠 cvsca 17230 mPwSer cmps 21830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-tset 17245 df-psr 21835 |
This theorem is referenced by: psrvscaval 21886 psrvscacl 21887 psrlmod 21896 psrass23l 21903 psrass23 21905 resspsrvsca 21913 mplvsca 21950 |
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