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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 25613. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 21658 the zero polynomial can be any degree (see mhp0cl 21671) 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 2733 | . . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | eqid 2733 | . . . . . . 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 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉) |
8 | mhpsclcl.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | 8 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
10 | mhpsclcl.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
11 | 10 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐶 ∈ 𝐾) |
12 | 1, 2, 3, 4, 5, 7, 9, 11 | mplascl 21607 | . . . . . 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 2737 | . . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (𝐼 × {0}) ↔ 𝑑 = (𝐼 × {0}))) | |
14 | 13 | ifbid 4550 | . . . . . . 7 ⊢ (𝑦 = 𝑑 → if(𝑦 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
15 | 14 | adantl 483 | . . . . . 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 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
17 | fvexd 6903 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
18 | 10, 17 | ifexd 4575 | . . . . . . 7 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
19 | 18 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
20 | 12, 15, 16, 19 | fvmptd 7001 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐴‘𝐶)‘𝑑) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
21 | 20 | neeq1d 3001 | . . . 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 4536 | . . . . . 6 ⊢ (¬ 𝑑 = (𝐼 × {0}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = (0g‘𝑅)) | |
23 | 22 | necon1ai 2969 | . . . . 5 ⊢ (if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅) → 𝑑 = (𝐼 × {0})) |
24 | fconstmpt 5736 | . . . . . . . 8 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0) | |
25 | 24 | oveq2i 7415 | . . . . . . 7 ⊢ ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) |
26 | nn0subm 20985 | . . . . . . . . 9 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
27 | eqid 2733 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
28 | 27 | submmnd 18690 | . . . . . . . . 9 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
29 | 26, 28 | ax-mp 5 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
30 | cnfld0 20954 | . . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld) | |
31 | 27, 30 | subm0 18692 | . . . . . . . . . 10 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
32 | 26, 31 | ax-mp 5 | . . . . . . . . 9 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
33 | 32 | gsumz 18713 | . . . . . . . 8 ⊢ (((ℂfld ↾s ℕ0) ∈ Mnd ∧ 𝐼 ∈ 𝑉) → ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) = 0) |
34 | 29, 7, 33 | sylancr 588 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) = 0) |
35 | 25, 34 | eqtrid 2785 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = 0) |
36 | oveq2 7412 | . . . . . . 7 ⊢ (𝑑 = (𝐼 × {0}) → ((ℂfld ↾s ℕ0) Σg 𝑑) = ((ℂfld ↾s ℕ0) Σg (𝐼 × {0}))) | |
37 | 36 | eqeq1d 2735 | . . . . . 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 3147 | . 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 2733 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
44 | 0nn0 12483 | . . . 4 ⊢ 0 ∈ ℕ0 | |
45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
46 | 1, 43, 4, 5, 6, 8 | mplasclf 21608 | . . . 4 ⊢ (𝜑 → 𝐴:𝐾⟶(Base‘𝑃)) |
47 | 46, 10 | ffvelcdmd 7083 | . . 3 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑃)) |
48 | 42, 1, 43, 3, 2, 6, 8, 45, 47 | ismhp3 21668 | . 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 257 | 1 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 Vcvv 3475 ifcif 4527 {csn 4627 ↦ cmpt 5230 × cxp 5673 ◡ccnv 5674 “ cima 5678 ‘cfv 6540 (class class class)co 7404 ↑m cmap 8816 Fincfn 8935 0cc0 11106 ℕcn 12208 ℕ0cn0 12468 Basecbs 17140 ↾s cress 17169 0gc0g 17381 Σg cgsu 17382 Mndcmnd 18621 SubMndcsubmnd 18666 Ringcrg 20047 ℂfldccnfld 20929 algSccascl 21391 mPoly cmpl 21441 mHomP cmhp 21654 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-ofr 7666 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-subrg 20349 df-lmod 20461 df-lss 20531 df-cnfld 20930 df-ascl 21394 df-psr 21444 df-mpl 21446 df-mhp 21658 |
This theorem is referenced by: mhppwdeg 21675 |
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