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Mirrors > Home > MPE Home > Th. List > crngbinom | Structured version Visualization version GIF version |
Description: The binomial theorem for commutative rings (special case of csrgbinom 19295): (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). (Contributed by AV, 24-Aug-2019.) |
Ref | Expression |
---|---|
crngbinom.s | ⊢ 𝑆 = (Base‘𝑅) |
crngbinom.m | ⊢ × = (.r‘𝑅) |
crngbinom.t | ⊢ · = (.g‘𝑅) |
crngbinom.a | ⊢ + = (+g‘𝑅) |
crngbinom.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
crngbinom.e | ⊢ ↑ = (.g‘𝐺) |
Ref | Expression |
---|---|
crngbinom | ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19307 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | ringsrg 19338 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑅 ∈ SRing) |
5 | crngbinom.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
6 | 5 | crngmgp 19304 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝐺 ∈ CMnd) |
8 | simpr 487 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
9 | 4, 7, 8 | 3jca 1124 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0)) |
10 | crngbinom.s | . . 3 ⊢ 𝑆 = (Base‘𝑅) | |
11 | crngbinom.m | . . 3 ⊢ × = (.r‘𝑅) | |
12 | crngbinom.t | . . 3 ⊢ · = (.g‘𝑅) | |
13 | crngbinom.a | . . 3 ⊢ + = (+g‘𝑅) | |
14 | crngbinom.e | . . 3 ⊢ ↑ = (.g‘𝐺) | |
15 | 10, 11, 12, 13, 5, 14 | csrgbinom 19295 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) |
16 | 9, 15 | sylan 582 | 1 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 0cc0 10536 − cmin 10869 ℕ0cn0 11896 ...cfz 12891 Ccbc 13661 Basecbs 16482 +gcplusg 16564 .rcmulr 16565 Σg cgsu 16713 .gcmg 18223 CMndccmn 18905 mulGrpcmgp 19238 SRingcsrg 19254 Ringcrg 19296 CRingccrg 19297 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-fac 13633 df-bc 13662 df-hash 13690 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-srg 19255 df-ring 19298 df-cring 19299 |
This theorem is referenced by: lply1binom 20473 freshmansdream 30859 |
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