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Theorem binom 11506
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 11505. 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 5896 . . . . . 6 |- ( x = 0 -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ 0 ) )
2 oveq2 5896 . . . . . . 7 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
3 oveq1 5895 . . . . . . . . 9 |- ( x = 0 -> ( x _C k ) = ( 0 _C k ) )
4 oveq1 5895 . . . . . . . . . . 11 |- ( x = 0 -> ( x - k ) = ( 0 - k ) )
54oveq2d 5904 . . . . . . . . . 10 |- ( x = 0 -> ( A ^ ( x - k ) ) = ( A ^ ( 0 - k ) ) )
65oveq1d 5903 . . . . . . . . 9 |- ( x = 0 -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) )
73, 6oveq12d 5906 . . . . . . . 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 11394 . . . . . 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 2202 . . . . 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 5896 . . . . . 6 |- ( x = n -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ n ) )
13 oveq2 5896 . . . . . . 7 |- ( x = n -> ( 0 ... x ) = ( 0 ... n ) )
14 oveq1 5895 . . . . . . . . 9 |- ( x = n -> ( x _C k ) = ( n _C k ) )
15 oveq1 5895 . . . . . . . . . . 11 |- ( x = n -> ( x - k ) = ( n - k ) )
1615oveq2d 5904 . . . . . . . . . 10 |- ( x = n -> ( A ^ ( x - k ) ) = ( A ^ ( n - k ) ) )
1716oveq1d 5903 . . . . . . . . 9 |- ( x = n -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) )
1814, 17oveq12d 5906 . . . . . . . 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 11394 . . . . . 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 2202 . . . . 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 5896 . . . . . 6 |- ( x = ( n + 1 ) -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ ( n + 1 ) ) )
24 oveq2 5896 . . . . . . 7 |- ( x = ( n + 1 ) -> ( 0 ... x ) = ( 0 ... ( n + 1 ) ) )
25 oveq1 5895 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( x _C k ) = ( ( n + 1 ) _C k ) )
26 oveq1 5895 . . . . . . . . . . 11 |- ( x = ( n + 1 ) -> ( x - k ) = ( ( n + 1 ) - k ) )
2726oveq2d 5904 . . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( A ^ ( x - k ) ) = ( A ^ ( ( n + 1 ) - k ) ) )
2827oveq1d 5903 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) )
2925, 28oveq12d 5906 . . . . . . . 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 11394 . . . . . 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 2202 . . . . 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 5896 . . . . . 6 |- ( x = N -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ N ) )
35 oveq2 5896 . . . . . . 7 |- ( x = N -> ( 0 ... x ) = ( 0 ... N ) )
36 oveq1 5895 . . . . . . . . 9 |- ( x = N -> ( x _C k ) = ( N _C k ) )
37 oveq1 5895 . . . . . . . . . . 11 |- ( x = N -> ( x - k ) = ( N - k ) )
3837oveq2d 5904 . . . . . . . . . 10 |- ( x = N -> ( A ^ ( x - k ) ) = ( A ^ ( N - k ) ) )
3938oveq1d 5903 . . . . . . . . 9 |- ( x = N -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) )
4036, 39oveq12d 5906 . . . . . . . 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 11394 . . . . . 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 2202 . . . . 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 10538 . . . . . . . . 9 |- ( A e. CC -> ( A ^ 0 ) = 1 )
46 exp0 10538 . . . . . . . . 9 |- ( B e. CC -> ( B ^ 0 ) = 1 )
4745, 46oveqan12d 5907 . . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
48 1t1e1 9085 . . . . . . . 8 |- ( 1 x. 1 ) = 1
4947, 48eqtrdi 2236 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
5049oveq2d 5904 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = ( 1 x. 1 ) )
5150, 48eqtrdi 2236 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = 1 )
52 0z 9278 . . . . . 6 |- 0 e. ZZ
53 ax-1cn 7918 . . . . . . 7 |- 1 e. CC
5451, 53eqeltrdi 2278 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC )
55 oveq2 5896 . . . . . . . . 9 |- ( k = 0 -> ( 0 _C k ) = ( 0 _C 0 ) )
56 0nn0 9205 . . . . . . . . . 10 |- 0 e. NN0
57 bcn0 10749 . . . . . . . . . 10 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
5856, 57ax-mp 5 . . . . . . . . 9 |- ( 0 _C 0 ) = 1
5955, 58eqtrdi 2236 . . . . . . . 8 |- ( k = 0 -> ( 0 _C k ) = 1 )
60 oveq2 5896 . . . . . . . . . . 11 |- ( k = 0 -> ( 0 - k ) = ( 0 - 0 ) )
61 0m0e0 9045 . . . . . . . . . . 11 |- ( 0 - 0 ) = 0
6260, 61eqtrdi 2236 . . . . . . . . . 10 |- ( k = 0 -> ( 0 - k ) = 0 )
6362oveq2d 5904 . . . . . . . . 9 |- ( k = 0 -> ( A ^ ( 0 - k ) ) = ( A ^ 0 ) )
64 oveq2 5896 . . . . . . . . 9 |- ( k = 0 -> ( B ^ k ) = ( B ^ 0 ) )
6563, 64oveq12d 5906 . . . . . . . 8 |- ( k = 0 -> ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
6659, 65oveq12d 5906 . . . . . . 7 |- ( k = 0 -> ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6766fsum1 11434 . . . . . 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 7950 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
7069exp0d 10662 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = 1 )
7151, 68, 703eqtr4rd 2231 . . . 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 11505 . . . . . 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 9381 . . 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 1195 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 979   = wceq 1363   e. wcel 2158  (class class class)co 5888   CCcc 7823   0cc0 7825   1c1 7826   + caddc 7828   x. cmul 7830   - cmin 8142   NN0cn0 9190   ZZcz 9267   ...cfz 10022   ^cexp 10533   _C cbc 10741   sum_csu 11375
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-fac 10720  df-bc 10742  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376
This theorem is referenced by:  binom1p  11507  efaddlem  11696
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