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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 2736 | . . 3 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | eqid 2736 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2736 | . . 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 21389 | . 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 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 4 | ringidcl 19989 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 6, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | 8, 3 | ring0cl 19990 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
12 | 6, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) |
13 | 10, 12 | ifcld 4532 | . . . . . 6 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅)) |
14 | 13 | ad2antrr 724 | . . . . 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 7063 | . . . 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 6855 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
17 | ovex 7390 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
18 | 17 | rabex 5289 | . . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V |
19 | 16, 18 | elmap 8809 | . . . 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 21346 | . . . 4 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) |
24 | 23 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) |
25 | 20, 24 | eleqtrrd 2841 | . 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 7064 | 1 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 ifcif 4486 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Fincfn 8883 0cc0 11051 1c1 11052 ℕcn 12153 ℕ0cn0 12413 Basecbs 17083 0gc0g 17321 1rcur 19913 Ringcrg 19964 mPwSer cmps 21306 mVar cmvr 21307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-tset 17152 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-mgp 19897 df-ur 19914 df-ring 19966 df-psr 21311 df-mvr 21312 |
This theorem is referenced by: mvrf1 21394 mvrcl2 21395 subrgmvrf 21435 mplbas2 21443 mvrf2 21468 evlseu 21493 |
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