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Mirrors > Home > MPE Home > Th. List > mvrcl | Structured version Visualization version GIF version |
Description: A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mvrcl.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mvrcl.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mvrcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mvrcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
mvrcl | ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2760 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mvrcl.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2760 | . . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | mvrcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mvrcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | mvrcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
7 | 1, 2, 3, 4, 5, 6 | mvrcl2 19648 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
8 | fvexd 6365 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ V) | |
9 | eqid 2760 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | eqid 2760 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
11 | 1, 9, 10, 3, 7 | psrelbas 19601 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | ffun 6209 | . . . 4 ⊢ ((𝑉‘𝑋):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅) → Fun (𝑉‘𝑋)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun (𝑉‘𝑋)) |
14 | fvexd 6365 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
15 | snfi 8205 | . . . 4 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin) |
17 | eqid 2760 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
18 | eqid 2760 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
19 | 4 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝐼 ∈ 𝑊) |
20 | 5 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑅 ∈ Ring) |
21 | 6 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑋 ∈ 𝐼) |
22 | simpr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) | |
23 | eldifsn 4462 | . . . . . . . 8 ⊢ (𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
24 | 22, 23 | sylib 208 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
25 | 24 | simpld 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
26 | 2, 10, 17, 18, 19, 20, 21, 25 | mvrval2 19644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅))) |
27 | 24 | simprd 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
28 | 27 | neneqd 2937 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ¬ 𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
29 | 28 | iffalsed 4241 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
30 | 26, 29 | eqtrd 2794 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = (0g‘𝑅)) |
31 | 11, 30 | suppss 7495 | . . 3 ⊢ (𝜑 → ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) |
32 | suppssfifsupp 8457 | . . 3 ⊢ ((((𝑉‘𝑋) ∈ V ∧ Fun (𝑉‘𝑋) ∧ (0g‘𝑅) ∈ V) ∧ ({(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin ∧ ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑉‘𝑋) finSupp (0g‘𝑅)) | |
33 | 8, 13, 14, 16, 31, 32 | syl32anc 1485 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) finSupp (0g‘𝑅)) |
34 | mvrcl.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
35 | mvrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
36 | 34, 1, 3, 17, 35 | mplelbas 19652 | . 2 ⊢ ((𝑉‘𝑋) ∈ 𝐵 ↔ ((𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑉‘𝑋) finSupp (0g‘𝑅))) |
37 | 7, 33, 36 | sylanbrc 701 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 {crab 3054 Vcvv 3340 ∖ cdif 3712 ⊆ wss 3715 ifcif 4230 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 ◡ccnv 5265 “ cima 5269 Fun wfun 6043 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 supp csupp 7464 ↑𝑚 cmap 8025 Fincfn 8123 finSupp cfsupp 8442 0cc0 10148 1c1 10149 ℕcn 11232 ℕ0cn0 11504 Basecbs 16079 0gc0g 16322 1rcur 18721 Ringcrg 18767 mPwSer cmps 19573 mVar cmvr 19574 mPoly cmpl 19575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-tset 16182 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-mgp 18710 df-ur 18722 df-ring 18769 df-psr 19578 df-mvr 19579 df-mpl 19580 |
This theorem is referenced by: subrgmvrf 19684 mplcoe3 19688 mplcoe5lem 19689 mplcoe5 19690 mplcoe2 19691 mplbas2 19692 mvrf2 19714 mpfproj 19753 mpfind 19758 vr1cl 19809 |
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