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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 26167. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 22158 the zero polynomial can be any degree (see mhp0cl 22168) 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 2735 | . . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | eqid 2735 | . . . . . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉) |
8 | mhpsclcl.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
9 | 8 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
10 | mhpsclcl.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐶 ∈ 𝐾) |
12 | 1, 2, 3, 4, 5, 7, 9, 11 | mplascl 22106 | . . . . . 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 2739 | . . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (𝐼 × {0}) ↔ 𝑑 = (𝐼 × {0}))) | |
14 | 13 | ifbid 4554 | . . . . . . 7 ⊢ (𝑦 = 𝑑 → if(𝑦 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
15 | 14 | adantl 481 | . . . . . 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 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
17 | fvexd 6922 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
18 | 10, 17 | ifexd 4579 | . . . . . . 7 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
19 | 18 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ∈ V) |
20 | 12, 15, 16, 19 | fvmptd 7023 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐴‘𝐶)‘𝑑) = if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅))) |
21 | 20 | neeq1d 2998 | . . . 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 4540 | . . . . . 6 ⊢ (¬ 𝑑 = (𝐼 × {0}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = (0g‘𝑅)) | |
23 | 22 | necon1ai 2966 | . . . . 5 ⊢ (if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅) → 𝑑 = (𝐼 × {0})) |
24 | fconstmpt 5751 | . . . . . . . 8 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0) | |
25 | 24 | oveq2i 7442 | . . . . . . 7 ⊢ ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = ((ℂfld ↾s ℕ0) Σg (𝑘 ∈ 𝐼 ↦ 0)) |
26 | nn0subm 21458 | . . . . . . . . 9 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
27 | eqid 2735 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
28 | 27 | submmnd 18839 | . . . . . . . . 9 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
29 | 26, 28 | ax-mp 5 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
30 | cnfld0 21423 | . . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld) | |
31 | 27, 30 | subm0 18841 | . . . . . . . . . 10 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
32 | 26, 31 | ax-mp 5 | . . . . . . . . 9 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
33 | 32 | gsumz 18862 | . . . . . . . 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 2787 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = 0) |
36 | oveq2 7439 | . . . . . . 7 ⊢ (𝑑 = (𝐼 × {0}) → ((ℂfld ↾s ℕ0) Σg 𝑑) = ((ℂfld ↾s ℕ0) Σg (𝐼 × {0}))) | |
37 | 36 | eqeq1d 2737 | . . . . . 6 ⊢ (𝑑 = (𝐼 × {0}) → (((ℂfld ↾s ℕ0) Σg 𝑑) = 0 ↔ ((ℂfld ↾s ℕ0) Σg (𝐼 × {0})) = 0)) |
38 | 35, 37 | syl5ibrcom 247 | . . . . 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 240 | . . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐴‘𝐶)‘𝑑) ≠ (0g‘𝑅) → ((ℂfld ↾s ℕ0) Σg 𝑑) = 0)) |
41 | 40 | ralrimiva 3144 | . 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 2735 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
44 | 0nn0 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
46 | 1, 43, 4, 5, 6, 8 | mplasclf 22107 | . . . 4 ⊢ (𝜑 → 𝐴:𝐾⟶(Base‘𝑃)) |
47 | 46, 10 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑃)) |
48 | 42, 1, 43, 3, 2, 45, 47 | ismhp3 22164 | . 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 Vcvv 3478 ifcif 4531 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Σg cgsu 17487 Mndcmnd 18760 SubMndcsubmnd 18808 Ringcrg 20251 ℂfldccnfld 21382 algSccascl 21890 mPoly cmpl 21944 mHomP cmhp 22151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-ascl 21893 df-psr 21947 df-mpl 21949 df-mhp 22158 |
This theorem is referenced by: mhppwdeg 22172 |
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