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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 26152. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 22140 the zero polynomial can be any degree (see mhp0cl 22150) 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 2737 | . . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
| 3 | eqid 2737 | . . . . . . 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 22088 | . . . . . 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 2741 | . . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (𝐼 × {0}) ↔ 𝑑 = (𝐼 × {0}))) | |
| 14 | 13 | ifbid 4549 | . . . . . . 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 6921 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
| 18 | 10, 17 | ifexd 4574 | . . . . . . 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 3000 | . . . 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 4534 | . . . . . 6 ⊢ (¬ 𝑑 = (𝐼 × {0}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) = (0g‘𝑅)) | |
| 23 | 22 | necon1ai 2968 | . . . . 5 ⊢ (if(𝑑 = (𝐼 × {0}), 𝐶, (0g‘𝑅)) ≠ (0g‘𝑅) → 𝑑 = (𝐼 × {0})) |
| 24 | fconstmpt 5747 | . . . . . . . 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 21440 | . . . . . . . . 9 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
| 27 | eqid 2737 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
| 28 | 27 | submmnd 18826 | . . . . . . . . 9 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
| 29 | 26, 28 | ax-mp 5 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
| 30 | cnfld0 21405 | . . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld) | |
| 31 | 27, 30 | subm0 18828 | . . . . . . . . . 10 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
| 32 | 26, 31 | ax-mp 5 | . . . . . . . . 9 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
| 33 | 32 | gsumz 18849 | . . . . . . . 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 2789 | . . . . . 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 2739 | . . . . . 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 3146 | . 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 2737 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 44 | 0nn0 12541 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
| 46 | 1, 43, 4, 5, 6, 8 | mplasclf 22089 | . . . 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 22146 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ifcif 4525 {csn 4626 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Σg cgsu 17485 Mndcmnd 18747 SubMndcsubmnd 18795 Ringcrg 20230 ℂfldccnfld 21364 algSccascl 21872 mPoly cmpl 21926 mHomP cmhp 22133 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-cnfld 21365 df-ascl 21875 df-psr 21929 df-mpl 21931 df-mhp 22140 |
| This theorem is referenced by: mhppwdeg 22154 |
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