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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 19237 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | cply1binom.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 20344 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | ringcmn 19260 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CMnd) |
6 | 5 | 3ad2ant1 1125 | . . . 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 20313 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ 𝐵) |
11 | 10 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
12 | simp3 1130 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
13 | cply1binom.a | . . . . 5 ⊢ + = (+g‘𝑃) | |
14 | 8, 13 | cmncom 18852 | . . . 4 ⊢ ((𝑃 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
15 | 6, 11, 12, 14 | syl3anc 1363 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
16 | 15 | oveq2d 7161 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑁 ↑ (𝐴 + 𝑋))) |
17 | 2 | ply1crng 20294 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
18 | 17 | 3ad2ant1 1125 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CRing) |
19 | simp2 1129 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
20 | 8 | eleq2i 2901 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑃)) |
21 | 20 | biimpi 217 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑃)) |
22 | 21 | 3ad2ant3 1127 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑃)) |
23 | 10, 8 | eleqtrdi 2920 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) |
24 | 23 | 3ad2ant1 1125 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
25 | eqid 2818 | . . . 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 19300 | . . 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 834 | . 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 2853 | 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 1079 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 0cc0 10525 − cmin 10858 ℕ0cn0 11885 ...cfz 12880 Ccbc 13650 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Σg cgsu 16702 .gcmg 18162 CMndccmn 18835 mulGrpcmgp 19168 Ringcrg 19226 CRingccrg 19227 var1cv1 20272 Poly1cpl1 20273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-fac 13622 df-bc 13651 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-subrg 19462 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-vr1 20277 df-ply1 20278 |
This theorem is referenced by: lply1binomsc 20403 |
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