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Mirrors > Home > MPE Home > Th. List > lply1binom | Structured version Visualization version GIF version |
Description: The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings: (𝑋 + 𝐴)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝑋↑𝑘)). (Contributed by AV, 25-Aug-2019.) |
Ref | Expression |
---|---|
cply1binom.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cply1binom.x | ⊢ 𝑋 = (var1‘𝑅) |
cply1binom.a | ⊢ + = (+g‘𝑃) |
cply1binom.m | ⊢ × = (.r‘𝑃) |
cply1binom.t | ⊢ · = (.g‘𝑃) |
cply1binom.g | ⊢ 𝐺 = (mulGrp‘𝑃) |
cply1binom.e | ⊢ ↑ = (.g‘𝐺) |
cply1binom.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
lply1binom | ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝑋)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19930 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | cply1binom.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 21571 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | ringcmn 19956 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CMnd) |
6 | 5 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CMnd) |
7 | cply1binom.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
8 | cply1binom.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
9 | 7, 2, 8 | vr1cl 21540 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ 𝐵) |
11 | 10 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
12 | simp3 1139 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
13 | cply1binom.a | . . . . 5 ⊢ + = (+g‘𝑃) | |
14 | 8, 13 | cmncom 19539 | . . . 4 ⊢ ((𝑃 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
15 | 6, 11, 12, 14 | syl3anc 1372 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
16 | 15 | oveq2d 7368 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑁 ↑ (𝐴 + 𝑋))) |
17 | 2 | ply1crng 21521 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
18 | 17 | 3ad2ant1 1134 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CRing) |
19 | simp2 1138 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
20 | 8 | eleq2i 2830 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑃)) |
21 | 20 | biimpi 215 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑃)) |
22 | 21 | 3ad2ant3 1136 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑃)) |
23 | 10, 8 | eleqtrdi 2849 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) |
24 | 23 | 3ad2ant1 1134 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
25 | eqid 2738 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
26 | cply1binom.m | . . . 4 ⊢ × = (.r‘𝑃) | |
27 | cply1binom.t | . . . 4 ⊢ · = (.g‘𝑃) | |
28 | cply1binom.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑃) | |
29 | cply1binom.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
30 | 25, 26, 27, 13, 28, 29 | crngbinom 20000 | . . 3 ⊢ (((𝑃 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Base‘𝑃) ∧ 𝑋 ∈ (Base‘𝑃))) → (𝑁 ↑ (𝐴 + 𝑋)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝑋)))))) |
31 | 18, 19, 22, 24, 30 | syl22anc 838 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝐴 + 𝑋)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝑋)))))) |
32 | 16, 31 | eqtrd 2778 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝑋)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7352 0cc0 11010 − cmin 11344 ℕ0cn0 12372 ...cfz 13379 Ccbc 14156 Basecbs 17043 +gcplusg 17093 .rcmulr 17094 Σg cgsu 17282 .gcmg 18831 CMndccmn 19521 mulGrpcmgp 19855 Ringcrg 19918 CRingccrg 19919 var1cv1 21499 Poly1cpl1 21500 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-ofr 7611 df-om 7796 df-1st 7914 df-2nd 7915 df-supp 8086 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-pm 8727 df-ixp 8795 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fsupp 9265 df-oi 9405 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-rp 12871 df-fz 13380 df-fzo 13523 df-seq 13862 df-fac 14128 df-bc 14157 df-hash 14185 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-tset 17112 df-ple 17113 df-0g 17283 df-gsum 17284 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cntz 19056 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-srg 19877 df-ring 19920 df-cring 19921 df-subrg 20173 df-psr 21264 df-mvr 21265 df-mpl 21266 df-opsr 21268 df-psr1 21503 df-vr1 21504 df-ply1 21505 |
This theorem is referenced by: lply1binomsc 21630 |
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