![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > binom11 | Structured version Visualization version GIF version |
Description: Special case of the binomial theorem for 2โ๐. (Contributed by Mario Carneiro, 13-Mar-2014.) |
Ref | Expression |
---|---|
binom11 | โข (๐ โ โ0 โ (2โ๐) = ฮฃ๐ โ (0...๐)(๐C๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 12313 | . . . 4 โข 2 = (1 + 1) | |
2 | 1 | oveq1i 7436 | . . 3 โข (2โ๐) = ((1 + 1)โ๐) |
3 | ax-1cn 11204 | . . . 4 โข 1 โ โ | |
4 | binom1p 15817 | . . . 4 โข ((1 โ โ โง ๐ โ โ0) โ ((1 + 1)โ๐) = ฮฃ๐ โ (0...๐)((๐C๐) ยท (1โ๐))) | |
5 | 3, 4 | mpan 688 | . . 3 โข (๐ โ โ0 โ ((1 + 1)โ๐) = ฮฃ๐ โ (0...๐)((๐C๐) ยท (1โ๐))) |
6 | 2, 5 | eqtrid 2780 | . 2 โข (๐ โ โ0 โ (2โ๐) = ฮฃ๐ โ (0...๐)((๐C๐) ยท (1โ๐))) |
7 | elfzelz 13541 | . . . . . 6 โข (๐ โ (0...๐) โ ๐ โ โค) | |
8 | 1exp 14096 | . . . . . 6 โข (๐ โ โค โ (1โ๐) = 1) | |
9 | 7, 8 | syl 17 | . . . . 5 โข (๐ โ (0...๐) โ (1โ๐) = 1) |
10 | 9 | oveq2d 7442 | . . . 4 โข (๐ โ (0...๐) โ ((๐C๐) ยท (1โ๐)) = ((๐C๐) ยท 1)) |
11 | bccl2 14322 | . . . . . 6 โข (๐ โ (0...๐) โ (๐C๐) โ โ) | |
12 | 11 | nncnd 12266 | . . . . 5 โข (๐ โ (0...๐) โ (๐C๐) โ โ) |
13 | 12 | mulridd 11269 | . . . 4 โข (๐ โ (0...๐) โ ((๐C๐) ยท 1) = (๐C๐)) |
14 | 10, 13 | eqtrd 2768 | . . 3 โข (๐ โ (0...๐) โ ((๐C๐) ยท (1โ๐)) = (๐C๐)) |
15 | 14 | sumeq2i 15685 | . 2 โข ฮฃ๐ โ (0...๐)((๐C๐) ยท (1โ๐)) = ฮฃ๐ โ (0...๐)(๐C๐) |
16 | 6, 15 | eqtrdi 2784 | 1 โข (๐ โ โ0 โ (2โ๐) = ฮฃ๐ โ (0...๐)(๐C๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 (class class class)co 7426 โcc 11144 0cc0 11146 1c1 11147 + caddc 11149 ยท cmul 11151 2c2 12305 โ0cn0 12510 โคcz 12596 ...cfz 13524 โcexp 14066 Ccbc 14301 ฮฃcsu 15672 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 |
This theorem is referenced by: chtublem 27164 lcmineqlem17 41548 |
Copyright terms: Public domain | W3C validator |