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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 4448 | . . . . . 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 2737 | . . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
4 | eqid 2737 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | mhpvarcl.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊) |
8 | mhpvarcl.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | 8 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
10 | mhpvarcl.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
11 | 10 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼) |
12 | simpr 488 | . . . . . . . 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 20947 | . . . . . . 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 2739 | . . . . . 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 2957 | . . . 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 20418 | . . . . . . 7 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
18 | eqid 2737 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
19 | cnfld0 20387 | . . . . . . . 8 ⊢ 0 = (0g‘ℂfld) | |
20 | 18, 19 | subm0 18242 | . . . . . . 7 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
21 | 17, 20 | ax-mp 5 | . . . . . 6 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
22 | 18 | submmnd 18240 | . . . . . . 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 2737 | . . . . . 6 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) | |
25 | 1nn0 12106 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
26 | 18 | submbas 18241 | . . . . . . . . 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 2836 | . . . . . . 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 19351 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1) |
31 | oveq2 7221 | . . . . . 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 2739 | . . . . 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 3105 | . 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 2737 | . . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
38 | eqid 2737 | . . 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 20977 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPoly 𝑅))) |
41 | 36, 37, 38, 4, 3, 6, 8, 39, 40 | ismhp3 21083 | . 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 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 {crab 3065 ifcif 4439 ↦ cmpt 5135 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 Fincfn 8626 0cc0 10729 1c1 10730 ℕcn 11830 ℕ0cn0 12090 Basecbs 16760 ↾s cress 16784 0gc0g 16944 Σg cgsu 16945 Mndcmnd 18173 SubMndcsubmnd 18217 1rcur 19516 Ringcrg 19562 ℂfldccnfld 20363 mVar cmvr 20864 mPoly cmpl 20865 mHomP cmhp 21069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-cnfld 20364 df-psr 20868 df-mvr 20869 df-mpl 20870 df-mhp 21073 |
This theorem is referenced by: (None) |
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