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Mirrors > Home > MPE Home > Th. List > mvrf | Structured version Visualization version GIF version |
Description: The power series variable function is a function from the index set to elements of the power series structure representing 𝑋𝑖 for each 𝑖. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
mvrf.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mvrf.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrf.b | ⊢ 𝐵 = (Base‘𝑆) |
mvrf.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrf.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
mvrf | ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrf.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
2 | eqid 2728 | . . 3 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | eqid 2728 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2728 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
5 | mvrf.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | mvrf.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | 1, 2, 3, 4, 5, 6 | mvrfval 21917 | . 2 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))))) |
8 | eqid 2728 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 4 | ringidcl 20196 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 6, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | 8, 3 | ring0cl 20197 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
12 | 6, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) |
13 | 10, 12 | ifcld 4571 | . . . . . 6 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
14 | 13 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
15 | 14 | fmpttd 7120 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
16 | fvex 6905 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
17 | ovex 7448 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
18 | 17 | rabex 5329 | . . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V |
19 | 16, 18 | elmap 8884 | . . . 4 ⊢ ((𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ↔ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
20 | 15, 19 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) |
21 | mvrf.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
22 | mvrf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
23 | 21, 8, 2, 22, 5 | psrbas 21872 | . . . 4 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) |
24 | 23 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) |
25 | 20, 24 | eleqtrrd 2832 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))) ∈ 𝐵) |
26 | 7, 25 | fmpt3d 7121 | 1 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3428 ifcif 4525 ↦ cmpt 5226 ◡ccnv 5672 “ cima 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 ↑m cmap 8839 Fincfn 8958 0cc0 11133 1c1 11134 ℕcn 12237 ℕ0cn0 12497 Basecbs 17174 0gc0g 17415 1rcur 20115 Ringcrg 20167 mPwSer cmps 21831 mVar cmvr 21832 |
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 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-tset 17246 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-mgp 20069 df-ur 20116 df-ring 20169 df-psr 21836 df-mvr 21837 |
This theorem is referenced by: mvrf1 21922 mvrcl2 21923 mvrf2 21929 subrgmvrf 21966 mplbas2 21974 evlseu 22023 |
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