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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 4534 | . . . . . 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 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊) |
| 8 | mhpvarcl.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
| 10 | mhpvarcl.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
| 11 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼) |
| 12 | simpr 484 | . . . . . . . 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 22003 | . . . . . . 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 | imbitrrid 246 | . . . . 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 21440 | . . . . . . 7 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
| 18 | eqid 2737 | . . . . . . . 8 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
| 19 | cnfld0 21405 | . . . . . . . 8 ⊢ 0 = (0g‘ℂfld) | |
| 20 | 18, 19 | subm0 18828 | . . . . . . 7 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂfld ↾s ℕ0))) |
| 21 | 17, 20 | ax-mp 5 | . . . . . 6 ⊢ 0 = (0g‘(ℂfld ↾s ℕ0)) |
| 22 | 18 | submmnd 18826 | . . . . . . 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 12542 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
| 26 | 18 | submbas 18827 | . . . . . . . . 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 2839 | . . . . . . 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 19984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((ℂfld ↾s ℕ0) Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1) |
| 31 | oveq2 7439 | . . . . . 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 247 | . . . 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 3146 | . 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 22012 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPoly 𝑅))) |
| 41 | 36, 37, 38, 4, 3, 39, 40 | ismhp3 22146 | . 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 257 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 ifcif 4525 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 0cc0 11155 1c1 11156 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Σg cgsu 17485 Mndcmnd 18747 SubMndcsubmnd 18795 1rcur 20178 Ringcrg 20230 ℂfldccnfld 21364 mVar cmvr 21925 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-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-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 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-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-cnfld 21365 df-psr 21929 df-mvr 21930 df-mpl 21931 df-mhp 22140 |
| This theorem is referenced by: (None) |
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