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Theorem binom 12011
Description: The binomial theorem: ( A + B ) ^ N is the sum from k = 0 to N of ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12010. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
binom |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
Distinct variable groups:   A, k   B, k   k, N
Proof of Theorem binom
Dummy variables n x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6015 . . . . . 6 |- ( x = 0 -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ 0 ) )
2 oveq2 6015 . . . . . . 7 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
3 oveq1 6014 . . . . . . . . 9 |- ( x = 0 -> ( x _C k ) = ( 0 _C k ) )
4 oveq1 6014 . . . . . . . . . . 11 |- ( x = 0 -> ( x - k ) = ( 0 - k ) )
54oveq2d 6023 . . . . . . . . . 10 |- ( x = 0 -> ( A ^ ( x - k ) ) = ( A ^ ( 0 - k ) ) )
65oveq1d 6022 . . . . . . . . 9 |- ( x = 0 -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) )
73, 6oveq12d 6025 . . . . . . . 8 |- ( x = 0 -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
87adantr 276 . . . . . . 7 |- ( ( x = 0 /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
92, 8sumeq12dv 11899 . . . . . 6 |- ( x = 0 -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
101, 9eqeq12d 2244 . . . . 5 |- ( x = 0 -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) ) )
1110imbi2d 230 . . . 4 |- ( x = 0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) ) ) )
12 oveq2 6015 . . . . . 6 |- ( x = n -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ n ) )
13 oveq2 6015 . . . . . . 7 |- ( x = n -> ( 0 ... x ) = ( 0 ... n ) )
14 oveq1 6014 . . . . . . . . 9 |- ( x = n -> ( x _C k ) = ( n _C k ) )
15 oveq1 6014 . . . . . . . . . . 11 |- ( x = n -> ( x - k ) = ( n - k ) )
1615oveq2d 6023 . . . . . . . . . 10 |- ( x = n -> ( A ^ ( x - k ) ) = ( A ^ ( n - k ) ) )
1716oveq1d 6022 . . . . . . . . 9 |- ( x = n -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) )
1814, 17oveq12d 6025 . . . . . . . 8 |- ( x = n -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
1918adantr 276 . . . . . . 7 |- ( ( x = n /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
2013, 19sumeq12dv 11899 . . . . . 6 |- ( x = n -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
2112, 20eqeq12d 2244 . . . . 5 |- ( x = n -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) )
2221imbi2d 230 . . . 4 |- ( x = n -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) ) )
23 oveq2 6015 . . . . . 6 |- ( x = ( n + 1 ) -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ ( n + 1 ) ) )
24 oveq2 6015 . . . . . . 7 |- ( x = ( n + 1 ) -> ( 0 ... x ) = ( 0 ... ( n + 1 ) ) )
25 oveq1 6014 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( x _C k ) = ( ( n + 1 ) _C k ) )
26 oveq1 6014 . . . . . . . . . . 11 |- ( x = ( n + 1 ) -> ( x - k ) = ( ( n + 1 ) - k ) )
2726oveq2d 6023 . . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( A ^ ( x - k ) ) = ( A ^ ( ( n + 1 ) - k ) ) )
2827oveq1d 6022 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) )
2925, 28oveq12d 6025 . . . . . . . 8 |- ( x = ( n + 1 ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
3029adantr 276 . . . . . . 7 |- ( ( x = ( n + 1 ) /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
3124, 30sumeq12dv 11899 . . . . . 6 |- ( x = ( n + 1 ) -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
3223, 31eqeq12d 2244 . . . . 5 |- ( x = ( n + 1 ) -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) )
3332imbi2d 230 . . . 4 |- ( x = ( n + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
34 oveq2 6015 . . . . . 6 |- ( x = N -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ N ) )
35 oveq2 6015 . . . . . . 7 |- ( x = N -> ( 0 ... x ) = ( 0 ... N ) )
36 oveq1 6014 . . . . . . . . 9 |- ( x = N -> ( x _C k ) = ( N _C k ) )
37 oveq1 6014 . . . . . . . . . . 11 |- ( x = N -> ( x - k ) = ( N - k ) )
3837oveq2d 6023 . . . . . . . . . 10 |- ( x = N -> ( A ^ ( x - k ) ) = ( A ^ ( N - k ) ) )
3938oveq1d 6022 . . . . . . . . 9 |- ( x = N -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) )
4036, 39oveq12d 6025 . . . . . . . 8 |- ( x = N -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
4140adantr 276 . . . . . . 7 |- ( ( x = N /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
4235, 41sumeq12dv 11899 . . . . . 6 |- ( x = N -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
4334, 42eqeq12d 2244 . . . . 5 |- ( x = N -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) ) )
4443imbi2d 230 . . . 4 |- ( x = N -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) ) ) )
45 exp0 10777 . . . . . . . . 9 |- ( A e. CC -> ( A ^ 0 ) = 1 )
46 exp0 10777 . . . . . . . . 9 |- ( B e. CC -> ( B ^ 0 ) = 1 )
4745, 46oveqan12d 6026 . . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
48 1t1e1 9274 . . . . . . . 8 |- ( 1 x. 1 ) = 1
4947, 48eqtrdi 2278 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
5049oveq2d 6023 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = ( 1 x. 1 ) )
5150, 48eqtrdi 2278 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = 1 )
52 0z 9468 . . . . . 6 |- 0 e. ZZ
53 ax-1cn 8103 . . . . . . 7 |- 1 e. CC
5451, 53eqeltrdi 2320 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC )
55 oveq2 6015 . . . . . . . . 9 |- ( k = 0 -> ( 0 _C k ) = ( 0 _C 0 ) )
56 0nn0 9395 . . . . . . . . . 10 |- 0 e. NN0
57 bcn0 10989 . . . . . . . . . 10 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
5856, 57ax-mp 5 . . . . . . . . 9 |- ( 0 _C 0 ) = 1
5955, 58eqtrdi 2278 . . . . . . . 8 |- ( k = 0 -> ( 0 _C k ) = 1 )
60 oveq2 6015 . . . . . . . . . . 11 |- ( k = 0 -> ( 0 - k ) = ( 0 - 0 ) )
61 0m0e0 9233 . . . . . . . . . . 11 |- ( 0 - 0 ) = 0
6260, 61eqtrdi 2278 . . . . . . . . . 10 |- ( k = 0 -> ( 0 - k ) = 0 )
6362oveq2d 6023 . . . . . . . . 9 |- ( k = 0 -> ( A ^ ( 0 - k ) ) = ( A ^ 0 ) )
64 oveq2 6015 . . . . . . . . 9 |- ( k = 0 -> ( B ^ k ) = ( B ^ 0 ) )
6563, 64oveq12d 6025 . . . . . . . 8 |- ( k = 0 -> ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
6659, 65oveq12d 6025 . . . . . . 7 |- ( k = 0 -> ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6766fsum1 11939 . . . . . 6 |- ( ( 0 e. ZZ /\ ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6852, 54, 67sylancr 414 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
69 addcl 8135 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
7069exp0d 10901 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = 1 )
7151, 68, 703eqtr4rd 2273 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
72 simprl 529 . . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> A e. CC )
73 simprr 531 . . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> B e. CC )
74 simpl 109 . . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> n e. NN0 )
75 id 19 . . . . . . 7 |- ( ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
7672, 73, 74, 75binomlem 12010 . . . . . 6 |- ( ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
7776exp31 364 . . . . 5 |- ( n e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
7877a2d 26 . . . 4 |- ( n e. NN0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) -> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
7911, 22, 33, 44, 71, 78nn0ind 9572 . . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) ) )
8079impcom 125 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
81803impa 1218 1 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   - cmin 8328   NN0cn0 9380   ZZcz 9457   ...cfz 10216   ^cexp 10772   _C cbc 10981   sum_csu 11880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-fac 10960  df-bc 10982  df-ihash 11010  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881
This theorem is referenced by:  binom1p  12012  efaddlem  12201
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