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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 19022): (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 19034 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | ringsrg 19065 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing) |
4 | 3 | adantr 473 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑅 ∈ SRing) |
5 | crngbinom.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
6 | 5 | crngmgp 19031 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝐺 ∈ CMnd) |
8 | simpr 477 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
9 | 4, 7, 8 | 3jca 1108 | . 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 19022 | . 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 572 | 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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ↦ cmpt 5009 ‘cfv 6190 (class class class)co 6978 0cc0 10337 − cmin 10672 ℕ0cn0 11710 ...cfz 12711 Ccbc 13480 Basecbs 16342 +gcplusg 16424 .rcmulr 16425 Σg cgsu 16573 .gcmg 18014 CMndccmn 18669 mulGrpcmgp 18965 SRingcsrg 18981 Ringcrg 19023 CRingccrg 19024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-oi 8771 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-fz 12712 df-fzo 12853 df-seq 13188 df-fac 13452 df-bc 13481 df-hash 13509 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-0g 16574 df-gsum 16575 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-submnd 17807 df-grp 17897 df-minusg 17898 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-srg 18982 df-ring 19025 df-cring 19026 |
This theorem is referenced by: lply1binom 20180 freshmansdream 30538 |
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