Theorem summodclem2 12127

Description: Lemma for summodc 12128. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
isummo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
summodclem2.g	|- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )

Assertion

Ref	Expression
summodclem2	|- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )

Distinct variable groups:   k, n, A   n, F   ph, k, n   A, f, j, m, k, n   B, n   f, F, k, m   ph, f, m   x, f, k, m, n   y, f, m

Allowed substitution hints:   ph( x, y, j)   A( x, y)   B( x, y, f, j, k, m)   F( x, y, j)   G( x, y, f, j, k, m, n)

Proof of Theorem summodclem2

Dummy variables a g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5690	. . . . 5 |- ( m = a -> ( ZZ>= ` m ) = ( ZZ>= ` a ) )
2	1	sseq2d 3278	. . . 4 |- ( m = a -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` a ) ) )
3	1	raleqdv 2755	. . . 4 |- ( m = a -> ( A. j e. ( ZZ>= ` m )DECID j e. A <-> A. j e. ( ZZ>= ` a )DECID j e. A ) )
4		seqeq1 10865	. . . . 5 |- ( m = a -> seq m ( + , F ) = seq a ( + , F ) )
5	4	breq1d 4135	. . . 4 |- ( m = a -> ( seq m ( + , F ) ~~> x <-> seq a ( + , F ) ~~> x ) )
6	2, 3, 5	3anbi123d 1353	. . 3 |- ( m = a -> ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) <-> ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) )
7	6	cbvrexv 2787	. 2 |- ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) <-> E. a e. ZZ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) )
8		simplr3 1072	. . . . . . . . 9 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq a ( + , F ) ~~> x )
9		simplr1 1070	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ( ZZ>= ` a ) )
10		uzssz 9921	. . . . . . . . . . . 12 |- ( ZZ>= ` a ) C_ ZZ
11	9, 10	sstrdi 3260	. . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ZZ )
12		1zzd 9650	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> 1 e. ZZ )
13		simprl 535	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> m e. NN )
14	13	nnzd 9746	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> m e. ZZ )
15	12, 14	fzfigd 10846	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) e. Fin )
16		simprr 537	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
17		f1oeng 7033	. . . . . . . . . . . . . 14 |- ( ( ( 1 ... m ) e. Fin /\ f : ( 1 ... m ) -1-1-onto-> A ) -> ( 1 ... m ) ~~ A )
18	15, 16, 17	syl2anc 415	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) ~~ A )
19	18	ensymd 7060	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A ~~ ( 1 ... m ) )
20		enfii 7166	. . . . . . . . . . . 12 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
21	15, 19, 20	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A e. Fin )
22		zfz1iso 11271	. . . . . . . . . . 11 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
23	11, 21, 22	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
24		isummo.1	. . . . . . . . . . . . 13 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
25		simplll 539	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> ph )
26		isummo.2	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. A ) -> B e. CC )
27	25, 26	sylan 283	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. A ) -> B e. CC )
28		eleq1w 2299	. . . . . . . . . . . . . . 15 |- ( j = k -> ( j e. A <-> k e. A ) )
29	28	dcbid 850	. . . . . . . . . . . . . 14 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
30		simpr2 1035	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) -> A. j e. ( ZZ>= ` a )DECID j e. A )
31	30	ad2antrr 492	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> A. j e. ( ZZ>= ` a )DECID j e. A )
32		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> k e. ( ZZ>= ` a ) )
33	29, 31, 32	rspcdva 2934	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> DECID k e. A )
34		summodclem2.g	. . . . . . . . . . . . 13 |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )
35		eqid 2238	. . . . . . . . . . . . 13 |- ( n e. NN |-> if ( n <_ m , [_ ( g ` n ) / k ]_ B , 0 ) ) = ( n e. NN |-> if ( n <_ m , [_ ( g ` n ) / k ]_ B , 0 ) )
36		simprll 543	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> m e. NN )
37		simpllr 540	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> a e. ZZ )
38		simplr1 1070	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> A C_ ( ZZ>= ` a ) )
39		simprlr 544	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
40		simprr 537	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 12126	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> seq a ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) )
42	41	expr 375	. . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) -> seq a ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) )
43	42	exlimdv 1872	. . . . . . . . . 10 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) -> seq a ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) )
44	23, 43	mpd 13	. . . . . . . . 9 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq a ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) )
45		climuni 12037	. . . . . . . . 9 |- ( ( seq a ( + , F ) ~~> x /\ seq a ( + , F ) ~~> ( seq 1 ( + , G ) ` m ) ) -> x = ( seq 1 ( + , G ) ` m ) )
46	8, 44, 45	syl2anc 415	. . . . . . . 8 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> x = ( seq 1 ( + , G ) ` m ) )
47	46	anassrs 404	. . . . . . 7 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> x = ( seq 1 ( + , G ) ` m ) )
48		eqeq2 2248	. . . . . . 7 |- ( y = ( seq 1 ( + , G ) ` m ) -> ( x = y <-> x = ( seq 1 ( + , G ) ` m ) ) )
49	47, 48	syl5ibrcom 157	. . . . . 6 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> ( y = ( seq 1 ( + , G ) ` m ) -> x = y ) )
50	49	expimpd 363	. . . . 5 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ m e. NN ) -> ( ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
51	50	exlimdv 1872	. . . 4 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) /\ m e. NN ) -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
52	51	rexlimdva 2668	. . 3 |- ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
53	52	r19.29an 2693	. 2 |- ( ( ph /\ E. a e. ZZ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
54	7, 53	sylan2b 287	1 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 846   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   [_csb 3147   C_ wss 3220   ifcif 3635   class class class wbr 4125   |-> cmpt 4187   -1-1-onto->wf1o 5371   ` cfv 5372   Isom wiso 5373  (class class class)co 6075   ~~ cen 7010   Fincfn 7012   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862  ♯chash 11192   ~~> cli 12022

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023

This theorem is referenced by:  summodc  12128