![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18767 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | cply1binom.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 19834 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | ringcmn 18790 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CMnd) |
6 | 5 | 3ad2ant1 1127 | . . . 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 19803 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ 𝐵) |
11 | 10 | 3ad2ant1 1127 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
12 | simp3 1132 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
13 | cply1binom.a | . . . . 5 ⊢ + = (+g‘𝑃) | |
14 | 8, 13 | cmncom 18417 | . . . 4 ⊢ ((𝑃 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
15 | 6, 11, 12, 14 | syl3anc 1476 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
16 | 15 | oveq2d 6810 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑁 ↑ (𝐴 + 𝑋))) |
17 | 2 | ply1crng 19784 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
18 | 17 | 3ad2ant1 1127 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CRing) |
19 | simp2 1131 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
20 | 8 | eleq2i 2842 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑃)) |
21 | 20 | biimpi 206 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑃)) |
22 | 21 | 3ad2ant3 1129 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑃)) |
23 | 10, 8 | syl6eleq 2860 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) |
24 | 23 | 3ad2ant1 1127 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
25 | eqid 2771 | . . . 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 18830 | . . 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 1477 | . 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 2805 | 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 1071 = wceq 1631 ∈ wcel 2145 ↦ cmpt 4864 ‘cfv 6032 (class class class)co 6794 0cc0 10139 − cmin 10469 ℕ0cn0 11495 ...cfz 12534 Ccbc 13294 Basecbs 16065 +gcplusg 16150 .rcmulr 16151 Σg cgsu 16310 .gcmg 17749 CMndccmn 18401 mulGrpcmgp 18698 Ringcrg 18756 CRingccrg 18757 var1cv1 19762 Poly1cpl1 19763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-isom 6041 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-of 7045 df-ofr 7046 df-om 7214 df-1st 7316 df-2nd 7317 df-supp 7448 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-2o 7715 df-oadd 7718 df-er 7897 df-map 8012 df-pm 8013 df-ixp 8064 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-fsupp 8433 df-oi 8572 df-card 8966 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-4 11284 df-5 11285 df-6 11286 df-7 11287 df-8 11288 df-9 11289 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-rp 12037 df-fz 12535 df-fzo 12675 df-seq 13010 df-fac 13266 df-bc 13295 df-hash 13323 df-struct 16067 df-ndx 16068 df-slot 16069 df-base 16071 df-sets 16072 df-ress 16073 df-plusg 16163 df-mulr 16164 df-sca 16166 df-vsca 16167 df-tset 16169 df-ple 16170 df-0g 16311 df-gsum 16312 df-mre 16455 df-mrc 16456 df-acs 16458 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-mhm 17544 df-submnd 17545 df-grp 17634 df-minusg 17635 df-mulg 17750 df-subg 17800 df-ghm 17867 df-cntz 17958 df-cmn 18403 df-abl 18404 df-mgp 18699 df-ur 18711 df-srg 18715 df-ring 18758 df-cring 18759 df-subrg 18989 df-psr 19572 df-mvr 19573 df-mpl 19574 df-opsr 19576 df-psr1 19766 df-vr1 19767 df-ply1 19768 |
This theorem is referenced by: lply1binomsc 19893 |
Copyright terms: Public domain | W3C validator |