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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 20263 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | cply1binom.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 22265 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | ringcmn 20296 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CMnd) |
6 | 5 | 3ad2ant1 1132 | . . . 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 22235 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ 𝐵) |
11 | 10 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
12 | simp3 1137 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
13 | cply1binom.a | . . . . 5 ⊢ + = (+g‘𝑃) | |
14 | 8, 13 | cmncom 19831 | . . . 4 ⊢ ((𝑃 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
15 | 6, 11, 12, 14 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑋 + 𝐴) = (𝐴 + 𝑋)) |
16 | 15 | oveq2d 7447 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑁 ↑ (𝐴 + 𝑋))) |
17 | 2 | ply1crng 22216 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
18 | 17 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑃 ∈ CRing) |
19 | simp2 1136 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
20 | 8 | eleq2i 2831 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑃)) |
21 | 20 | biimpi 216 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑃)) |
22 | 21 | 3ad2ant3 1134 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑃)) |
23 | 10, 8 | eleqtrdi 2849 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃)) |
24 | 23 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃)) |
25 | eqid 2735 | . . . 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 20349 | . . 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 839 | . 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 2775 | 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 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 0cc0 11153 − cmin 11490 ℕ0cn0 12524 ...cfz 13544 Ccbc 14338 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Σg cgsu 17487 .gcmg 19098 CMndccmn 19813 mulGrpcmgp 20152 Ringcrg 20251 CRingccrg 20252 var1cv1 22193 Poly1cpl1 22194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-fac 14310 df-bc 14339 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 |
This theorem is referenced by: lply1binomsc 22331 |
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