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Mirrors > Home > MPE Home > Th. List > mhpvarcl | Structured version Visualization version GIF version |
Description: A power series variable is a polynomial of degree 1. (Contributed by SN, 25-May-2024.) |
Ref | Expression |
---|---|
mhpvarcl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpvarcl.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mhpvarcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mhpvarcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mhpvarcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
mhpvarcl | ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iffalse 4422 | . . . . . 6 ⊢ (¬ 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) | |
2 | mhpvarcl.v | . . . . . . . 8 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2759 | . . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
4 | eqid 2759 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2759 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | mhpvarcl.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | adantr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊) |
8 | mhpvarcl.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | 8 | adantr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
10 | mhpvarcl.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
11 | 10 | adantr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼) |
12 | simpr 489 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
13 | 2, 3, 4, 5, 7, 9, 11, 12 | mvrval2 20735 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑉‘𝑋)‘𝑑) = if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅))) |
14 | 13 | eqeq1d 2761 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) = (0g‘𝑅) ↔ if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))) |
15 | 1, 14 | syl5ibr 249 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (¬ 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((𝑉‘𝑋)‘𝑑) = (0g‘𝑅))) |
16 | 15 | necon1ad 2966 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) ≠ (0g‘𝑅) → 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
17 | nn0subm 20206 | . . . . . . 7 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
18 | eqid 2759 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
19 | cnfld0 20175 | . . . . . . . 8 ⊢ 0 = (0g‘ℂfld) | |
20 | 18, 19 | subm0 18031 | . . . . . . 7 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
21 | 17, 20 | ax-mp 5 | . . . . . 6 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
22 | 18 | submmnd 18029 | . . . . . . 7 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
23 | 17, 22 | mp1i 13 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (ℂfld ↾s ℕ0) ∈ Mnd) |
24 | eqid 2759 | . . . . . 6 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) | |
25 | 1nn0 11935 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
26 | 18 | submbas 18030 | . . . . . . . . 9 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → ℕ0 = (Base‘(ℂfld ↾s ℕ0))) |
27 | 17, 26 | ax-mp 5 | . . . . . . . 8 ⊢ ℕ0 = (Base‘(ℂfld ↾s ℕ0)) |
28 | 25, 27 | eleqtri 2849 | . . . . . . 7 ⊢ 1 ∈ (Base‘(ℂfld ↾s ℕ0)) |
29 | 28 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 1 ∈ (Base‘(ℂfld ↾s ℕ0))) |
30 | 21, 23, 7, 11, 24, 29 | gsummptif1n0 19139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1) |
31 | oveq2 7151 | . . . . . 6 ⊢ (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((ℂfld ↾s ℕ0) Σg 𝑑) = ((ℂfld ↾s ℕ0) Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
32 | 31 | eqeq1d 2761 | . . . . 5 ⊢ (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (((ℂfld ↾s ℕ0) Σg 𝑑) = 1 ↔ ((ℂfld ↾s ℕ0) Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1)) |
33 | 30, 32 | syl5ibrcom 250 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 1)) |
34 | 16, 33 | syld 47 | . . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 1)) |
35 | 34 | ralrimiva 3111 | . 2 ⊢ (𝜑 → ∀𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝑉‘𝑋)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 1)) |
36 | mhpvarcl.h | . . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
37 | eqid 2759 | . . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
38 | eqid 2759 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
39 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ ℕ0) |
40 | 37, 2, 38, 6, 8, 10 | mvrcl 20765 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPoly 𝑅))) |
41 | 36, 37, 38, 4, 3, 6, 8, 39, 40 | ismhp3 20871 | . 2 ⊢ (𝜑 → ((𝑉‘𝑋) ∈ (𝐻‘1) ↔ ∀𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝑉‘𝑋)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 1))) |
42 | 35, 41 | mpbird 260 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 ∀wral 3068 {crab 3072 ifcif 4413 ↦ cmpt 5105 ◡ccnv 5516 “ cima 5520 ‘cfv 6328 (class class class)co 7143 ↑m cmap 8409 Fincfn 8520 0cc0 10560 1c1 10561 ℕcn 11659 ℕ0cn0 11919 Basecbs 16526 ↾s cress 16527 0gc0g 16756 Σg cgsu 16757 Mndcmnd 17962 SubMndcsubmnd 18006 1rcur 19304 Ringcrg 19350 ℂfldccnfld 20151 mVar cmvr 20652 mPoly cmpl 20653 mHomP cmhp 20857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-addf 10639 ax-mulf 10640 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-fzo 13068 df-seq 13404 df-hash 13726 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-sca 16624 df-vsca 16625 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-0g 16758 df-gsum 16759 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-grp 18157 df-mulg 18277 df-cntz 18499 df-cmn 18960 df-mgp 19293 df-ur 19305 df-ring 19352 df-cring 19353 df-cnfld 20152 df-psr 20656 df-mvr 20657 df-mpl 20658 df-mhp 20861 |
This theorem is referenced by: (None) |
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