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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 20137): (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 20150 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | ringsrg 20196 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑅 ∈ SRing) |
5 | crngbinom.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
6 | 5 | crngmgp 20146 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝐺 ∈ CMnd) |
8 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
9 | 4, 7, 8 | 3jca 1125 | . 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 20137 | . 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 579 | 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 0cc0 11112 − cmin 11448 ℕ0cn0 12476 ...cfz 13490 Ccbc 14267 Basecbs 17153 +gcplusg 17206 .rcmulr 17207 Σg cgsu 17395 .gcmg 18995 CMndccmn 19700 mulGrpcmgp 20039 SRingcsrg 20091 Ringcrg 20138 CRingccrg 20139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-fac 14239 df-bc 14268 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 |
This theorem is referenced by: freshmansdream 21469 lply1binom 22184 |
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