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Mirrors > Home > MPE Home > Th. List > mhpsclcl | Structured version Visualization version GIF version |
Description: A scalar (or constant) polynomial has degree 0. Compare deg1scl 25478. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 21523 the zero polynomial can be any degree (see mhp0cl 21536) so there is no exception. (Contributed by SN, 25-May-2024.) |
Ref | Expression |
---|---|
mhpsclcl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpsclcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpsclcl.a | ⊢ 𝐴 = (algSc‘𝑃) |
mhpsclcl.k | ⊢ 𝐾 = (Base‘𝑅) |
mhpsclcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpsclcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mhpsclcl.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
Ref | Expression |
---|---|
mhpsclcl | ⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpsclcl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | eqid 2736 | . . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | eqid 2736 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | mhpsclcl.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
5 | mhpsclcl.a | . . . . . . 7 ⊢ 𝐴 = (algSc‘𝑃) | |
6 | mhpsclcl.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉) |
8 | mhpsclcl.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | 8 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
10 | mhpsclcl.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
11 | 10 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐶 ∈ 𝐾) |
12 | 1, 2, 3, 4, 5, 7, 9, 11 | mplascl 21472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐴‘𝐶) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝐶, (0g‘𝑅)))) |
13 | eqeq1 2740 | . . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (𝐼 × {0}) ↔ 𝑑 = (𝐼 × {0}))) | |
14 | 13 | ifbid 4509 | . . . . . . 7 ⊢ (𝑦 = 𝑑 → if(𝑦 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
15 | 14 | adantl 482 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑦 = 𝑑) → if(𝑦 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
16 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
17 | fvexd 6857 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
18 | 10, 17 | ifexd 4534 | . . . . . . 7 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
19 | 18 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
20 | 12, 15, 16, 19 | fvmptd 6955 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐴‘𝐶)‘𝑑) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
21 | 20 | neeq1d 3003 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐴‘𝐶)‘𝑑) ≠ (0g‘𝑅) ↔ if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅))) |
22 | iffalse 4495 | . . . . . 6 ⊢ (¬ 𝑑 = (𝐼 × {0}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = (0g‘𝑅)) | |
23 | 22 | necon1ai 2971 | . . . . 5 ⊢ (if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅) → 𝑑 = (𝐼 × {0})) |
24 | fconstmpt 5694 | . . . . . . . 8 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0) | |
25 | 24 | oveq2i 7368 | . . . . . . 7 ⊢ ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) |
26 | nn0subm 20852 | . . . . . . . . 9 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
27 | eqid 2736 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
28 | 27 | submmnd 18624 | . . . . . . . . 9 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
29 | 26, 28 | ax-mp 5 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
30 | cnfld0 20821 | . . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld) | |
31 | 27, 30 | subm0 18626 | . . . . . . . . . 10 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
32 | 26, 31 | ax-mp 5 | . . . . . . . . 9 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
33 | 32 | gsumz 18646 | . . . . . . . 8 ⊢ (((ℂfld ↾s ℕ0) ∈ Mnd ∧ 𝐼 ∈ 𝑉) → ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) = 0) |
34 | 29, 7, 33 | sylancr 587 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) = 0) |
35 | 25, 34 | eqtrid 2788 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = 0) |
36 | oveq2 7365 | . . . . . . 7 ⊢ (𝑑 = (𝐼 × {0}) → ((ℂfld ↾s ℕ0) Σg 𝑑) = ((ℂfld ↾s ℕ0) Σg (𝐼 × {0}))) | |
37 | 36 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑑 = (𝐼 × {0}) → (((ℂfld ↾s ℕ0) Σg 𝑑) = 0 ↔ ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = 0)) |
38 | 35, 37 | syl5ibrcom 246 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 = (𝐼 × {0}) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0)) |
39 | 23, 38 | syl5 34 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0)) |
40 | 21, 39 | sylbid 239 | . . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐴‘𝐶)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0)) |
41 | 40 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝐴‘𝐶)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0)) |
42 | mhpsclcl.h | . . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
43 | eqid 2736 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
44 | 0nn0 12428 | . . . 4 ⊢ 0 ∈ ℕ0 | |
45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
46 | 1, 43, 4, 5, 6, 8 | mplasclf 21473 | . . . 4 ⊢ (𝜑 → 𝐴:𝐾⟶(Base‘𝑃)) |
47 | 46, 10 | ffvelcdmd 7036 | . . 3 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑃)) |
48 | 42, 1, 43, 3, 2, 6, 8, 45, 47 | ismhp3 21533 | . 2 ⊢ (𝜑 → ((𝐴‘𝐶) ∈ (𝐻‘0) ↔ ∀𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (((𝐴‘𝐶)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0))) |
49 | 41, 48 | mpbird 256 | 1 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3407 Vcvv 3445 ifcif 4486 {csn 4586 ↦ cmpt 5188 × cxp 5631 ◡ccnv 5632 “ cima 5636 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Fincfn 8883 0cc0 11051 ℕcn 12153 ℕ0cn0 12413 Basecbs 17083 ↾s cress 17112 0gc0g 17321 Σg cgsu 17322 Mndcmnd 18556 SubMndcsubmnd 18600 Ringcrg 19964 ℂfldccnfld 20796 algSccascl 21258 mPoly cmpl 21308 mHomP cmhp 21519 |
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 ax-addf 11130 ax-mulf 11131 |
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-int 4908 df-iun 4956 df-iin 4957 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-se 5589 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-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 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-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 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-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-lmod 20324 df-lss 20393 df-cnfld 20797 df-ascl 21261 df-psr 21311 df-mpl 21313 df-mhp 21523 |
This theorem is referenced by: mhppwdeg 21540 |
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