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Mirrors > Home > MPE Home > Th. List > csrgbinom | Structured version Visualization version GIF version |
Description: The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.) |
Ref | Expression |
---|---|
srgbinom.s | ⊢ 𝑆 = (Base‘𝑅) |
srgbinom.m | ⊢ × = (.r‘𝑅) |
srgbinom.t | ⊢ · = (.g‘𝑅) |
srgbinom.a | ⊢ + = (+g‘𝑅) |
srgbinom.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
srgbinom.e | ⊢ ↑ = (.g‘𝐺) |
Ref | Expression |
---|---|
csrgbinom | ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 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 | 3simpb 1129 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) → (𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0)) | |
2 | 1 | adantr 473 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0)) |
3 | simprl 758 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
4 | simprr 760 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
5 | simpl2 1172 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐺 ∈ CMnd) | |
6 | srgbinom.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
7 | srgbinom.s | . . . . 5 ⊢ 𝑆 = (Base‘𝑅) | |
8 | 6, 7 | mgpbas 18962 | . . . 4 ⊢ 𝑆 = (Base‘𝐺) |
9 | srgbinom.m | . . . . 5 ⊢ × = (.r‘𝑅) | |
10 | 6, 9 | mgpplusg 18960 | . . . 4 ⊢ × = (+g‘𝐺) |
11 | 8, 10 | cmncom 18676 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 × 𝐵) = (𝐵 × 𝐴)) |
12 | 5, 3, 4, 11 | syl3anc 1351 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) = (𝐵 × 𝐴)) |
13 | srgbinom.t | . . 3 ⊢ · = (.g‘𝑅) | |
14 | srgbinom.a | . . 3 ⊢ + = (+g‘𝑅) | |
15 | srgbinom.e | . . 3 ⊢ ↑ = (.g‘𝐺) | |
16 | 7, 9, 13, 14, 6, 15 | srgbinom 19012 | . 2 ⊢ (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) |
17 | 2, 3, 4, 12, 16 | syl13anc 1352 | 1 ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 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 5002 ‘cfv 6182 (class class class)co 6970 0cc0 10329 − cmin 10664 ℕ0cn0 11701 ...cfz 12702 Ccbc 13471 Basecbs 16333 +gcplusg 16415 .rcmulr 16416 Σg cgsu 16564 .gcmg 18005 CMndccmn 18660 mulGrpcmgp 18956 SRingcsrg 18972 |
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 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
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 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-oi 8763 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-rp 12199 df-fz 12703 df-fzo 12844 df-seq 13179 df-fac 13443 df-bc 13472 df-hash 13500 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-0g 16565 df-gsum 16566 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-mhm 17797 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-mgp 18957 df-ur 18969 df-srg 18973 |
This theorem is referenced by: crngbinom 19088 |
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