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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 20243 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 2 | cply1binom.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 3 | 2 | ply1ring 22250 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) | 
| 4 | ringcmn 20280 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
| 5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CMnd) | 
| 6 | 5 | 3ad2ant1 1133 | . . . 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 22220 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) | 
| 10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ 𝐵) | 
| 11 | 10 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ 𝐵) | 
| 12 | simp3 1138 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
| 13 | cply1binom.a | . . . . 5 ⊢ + = (+g‘𝑃) | |
| 14 | 8, 13 | cmncom 19817 | . . . 4 ⊢ ((𝑃 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) | 
| 15 | 6, 11, 12, 14 | syl3anc 1372 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) | 
| 16 | 15 | oveq2d 7448 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑁 ↑ (𝐴 + 𝑋))) | 
| 17 | 2 | ply1crng 22201 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) | 
| 18 | 17 | 3ad2ant1 1133 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CRing) | 
| 19 | simp2 1137 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
| 20 | 8 | eleq2i 2832 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑃)) | 
| 21 | 20 | biimpi 216 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑃)) | 
| 22 | 21 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑃)) | 
| 23 | 10, 8 | eleqtrdi 2850 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) | 
| 24 | 23 | 3ad2ant1 1133 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) | 
| 25 | eqid 2736 | . . . 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 20333 | . . 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 838 | . 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 2776 | 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 1086 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 0cc0 11156 − cmin 11493 ℕ0cn0 12528 ...cfz 13548 Ccbc 14342 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Σg cgsu 17486 .gcmg 19086 CMndccmn 19799 mulGrpcmgp 20138 Ringcrg 20231 CRingccrg 20232 var1cv1 22178 Poly1cpl1 22179 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-seq 14044 df-fac 14314 df-bc 14343 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-subrng 20547 df-subrg 20571 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-psr1 22182 df-vr1 22183 df-ply1 22184 | 
| This theorem is referenced by: lply1binomsc 22316 | 
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