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Theorem binom 11285
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 11284. 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 5790 . . . . . 6 |- ( x = 0 -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ 0 ) )
2 oveq2 5790 . . . . . . 7 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
3 oveq1 5789 . . . . . . . . 9 |- ( x = 0 -> ( x _C k ) = ( 0 _C k ) )
4 oveq1 5789 . . . . . . . . . . 11 |- ( x = 0 -> ( x - k ) = ( 0 - k ) )
54oveq2d 5798 . . . . . . . . . 10 |- ( x = 0 -> ( A ^ ( x - k ) ) = ( A ^ ( 0 - k ) ) )
65oveq1d 5797 . . . . . . . . 9 |- ( x = 0 -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) )
73, 6oveq12d 5800 . . . . . . . 8 |- ( x = 0 -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
87adantr 274 . . . . . . 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 11173 . . . . . 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 2155 . . . . 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 229 . . . 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 5790 . . . . . 6 |- ( x = n -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ n ) )
13 oveq2 5790 . . . . . . 7 |- ( x = n -> ( 0 ... x ) = ( 0 ... n ) )
14 oveq1 5789 . . . . . . . . 9 |- ( x = n -> ( x _C k ) = ( n _C k ) )
15 oveq1 5789 . . . . . . . . . . 11 |- ( x = n -> ( x - k ) = ( n - k ) )
1615oveq2d 5798 . . . . . . . . . 10 |- ( x = n -> ( A ^ ( x - k ) ) = ( A ^ ( n - k ) ) )
1716oveq1d 5797 . . . . . . . . 9 |- ( x = n -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) )
1814, 17oveq12d 5800 . . . . . . . 8 |- ( x = n -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
1918adantr 274 . . . . . . 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 11173 . . . . . 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 2155 . . . . 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 229 . . . 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 5790 . . . . . 6 |- ( x = ( n + 1 ) -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ ( n + 1 ) ) )
24 oveq2 5790 . . . . . . 7 |- ( x = ( n + 1 ) -> ( 0 ... x ) = ( 0 ... ( n + 1 ) ) )
25 oveq1 5789 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( x _C k ) = ( ( n + 1 ) _C k ) )
26 oveq1 5789 . . . . . . . . . . 11 |- ( x = ( n + 1 ) -> ( x - k ) = ( ( n + 1 ) - k ) )
2726oveq2d 5798 . . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( A ^ ( x - k ) ) = ( A ^ ( ( n + 1 ) - k ) ) )
2827oveq1d 5797 . . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) )
2925, 28oveq12d 5800 . . . . . . . 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 274 . . . . . . 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 11173 . . . . . 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 2155 . . . . 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 229 . . . 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 5790 . . . . . 6 |- ( x = N -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ N ) )
35 oveq2 5790 . . . . . . 7 |- ( x = N -> ( 0 ... x ) = ( 0 ... N ) )
36 oveq1 5789 . . . . . . . . 9 |- ( x = N -> ( x _C k ) = ( N _C k ) )
37 oveq1 5789 . . . . . . . . . . 11 |- ( x = N -> ( x - k ) = ( N - k ) )
3837oveq2d 5798 . . . . . . . . . 10 |- ( x = N -> ( A ^ ( x - k ) ) = ( A ^ ( N - k ) ) )
3938oveq1d 5797 . . . . . . . . 9 |- ( x = N -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) )
4036, 39oveq12d 5800 . . . . . . . 8 |- ( x = N -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
4140adantr 274 . . . . . . 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 11173 . . . . . 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 2155 . . . . 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 229 . . . 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 10328 . . . . . . . . 9 |- ( A e. CC -> ( A ^ 0 ) = 1 )
46 exp0 10328 . . . . . . . . 9 |- ( B e. CC -> ( B ^ 0 ) = 1 )
4745, 46oveqan12d 5801 . . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
48 1t1e1 8896 . . . . . . . 8 |- ( 1 x. 1 ) = 1
4947, 48eqtrdi 2189 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
5049oveq2d 5798 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = ( 1 x. 1 ) )
5150, 48eqtrdi 2189 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = 1 )
52 0z 9089 . . . . . 6 |- 0 e. ZZ
53 ax-1cn 7737 . . . . . . 7 |- 1 e. CC
5451, 53eqeltrdi 2231 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC )
55 oveq2 5790 . . . . . . . . 9 |- ( k = 0 -> ( 0 _C k ) = ( 0 _C 0 ) )
56 0nn0 9016 . . . . . . . . . 10 |- 0 e. NN0
57 bcn0 10533 . . . . . . . . . 10 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
5856, 57ax-mp 5 . . . . . . . . 9 |- ( 0 _C 0 ) = 1
5955, 58eqtrdi 2189 . . . . . . . 8 |- ( k = 0 -> ( 0 _C k ) = 1 )
60 oveq2 5790 . . . . . . . . . . 11 |- ( k = 0 -> ( 0 - k ) = ( 0 - 0 ) )
61 0m0e0 8856 . . . . . . . . . . 11 |- ( 0 - 0 ) = 0
6260, 61eqtrdi 2189 . . . . . . . . . 10 |- ( k = 0 -> ( 0 - k ) = 0 )
6362oveq2d 5798 . . . . . . . . 9 |- ( k = 0 -> ( A ^ ( 0 - k ) ) = ( A ^ 0 ) )
64 oveq2 5790 . . . . . . . . 9 |- ( k = 0 -> ( B ^ k ) = ( B ^ 0 ) )
6563, 64oveq12d 5800 . . . . . . . 8 |- ( k = 0 -> ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
6659, 65oveq12d 5800 . . . . . . 7 |- ( k = 0 -> ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6766fsum1 11213 . . . . . 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 411 . . . . 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 7769 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
7069exp0d 10449 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = 1 )
7151, 68, 703eqtr4rd 2184 . . . 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 521 . . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> A e. CC )
73 simprr 522 . . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> B e. CC )
74 simpl 108 . . . . . . 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 11284 . . . . . 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 362 . . . . 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 9189 . . 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 124 . 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 1177 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 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   - cmin 7957   NN0cn0 9001   ZZcz 9078   ...cfz 9821   ^cexp 10323   _C cbc 10525   sum_csu 11154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155
This theorem is referenced by:  binom1p  11286  efaddlem  11417
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