Theorem cvmliftlem5 31271

Description: Lemma for cvmlift 31281. Definition of Q at a successor. This is a function defined on W as `' ( T |` I ) o. G where I is the unique covering set of 2nd ` ( T ` M ) that contains Q ( M - 1 ) evaluated at the last defined point, namely ( M - 1 ) / N (note that for M = 1 this is using the seed value Q ( 0 ) ( 0 ) = P). (Contributed by Mario Carneiro, 15-Feb-2015.)

Hypotheses

Ref	Expression
cvmliftlem.1	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
cvmliftlem.b	|- B = U. C
cvmliftlem.x	|- X = U. J
cvmliftlem.f	|- ( ph -> F e. ( C CovMap J ) )
cvmliftlem.g	|- ( ph -> G e. ( II Cn J ) )
cvmliftlem.p	|- ( ph -> P e. B )
cvmliftlem.e	|- ( ph -> ( F ` P ) = ( G ` 0 ) )
cvmliftlem.n	|- ( ph -> N e. NN )
cvmliftlem.t	|- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
cvmliftlem.a	|- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
cvmliftlem.l	|- L = ( topGen ` ran (,) )
cvmliftlem.q	|- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
cvmliftlem5.3	|- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )

Assertion

Ref	Expression
cvmliftlem5	|- ( ( ph /\ M e. NN ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )

Distinct variable groups:   v, b, z, B   j, b, k, m, s, u, x, F, v, z   z, L   M, b, j, k, m, s, u, v, x, z   P, b, k, m, u, v, x, z   C, b, j, k, s, u, v, z   ph, j, s, x, z   N, b, k, m, u, v, x, z   S, b, j, k, s, u, v, x, z   j, X   G, b, j, k, m, s, u, v, x, z   T, b, j, k, m, s, u, v, x, z   J, b, j, k, s, u, v, x, z   Q, b, k, m, u, v, x, z   k, W, m, x, z

Allowed substitution hints:   ph( v, u, k, m, b)   B( x, u, j, k, m, s)   C( x, m)   P( j, s)   Q( j, s)   S( m)   J( m)   L( x, v, u, j, k, m, s, b)   N( j, s)   W( v, u, j, s, b)   X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem5

Step	Hyp	Ref	Expression
1		0z 11388	. . . 4 |- 0 e. ZZ
2		simpr 477	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN )
3		nnuz 11723	. . . . . 6 |- NN = ( ZZ>= ` 1 )
4		1e0p1 11552	. . . . . . 7 |- 1 = ( 0 + 1 )
5	4	fveq2i 6194	. . . . . 6 |- ( ZZ>= ` 1 ) = ( ZZ>= ` ( 0 + 1 ) )
6	3, 5	eqtri 2644	. . . . 5 |- NN = ( ZZ>= ` ( 0 + 1 ) )
7	2, 6	syl6eleq 2711	. . . 4 |- ( ( ph /\ M e. NN ) -> M e. ( ZZ>= ` ( 0 + 1 ) ) )
8		seqm1 12818	. . . 4 |- ( ( 0 e. ZZ /\ M e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` M ) = ( ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) ) )
9	1, 7, 8	sylancr 695	. . 3 |- ( ( ph /\ M e. NN ) -> ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` M ) = ( ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) ) )
10		cvmliftlem.q	. . . 4 |- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
11	10	fveq1i 6192	. . 3 |- ( Q ` M ) = ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` M )
12	10	fveq1i 6192	. . . 4 |- ( Q ` ( M - 1 ) ) = ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` ( M - 1 ) )
13	12	oveq1i 6660	. . 3 |- ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) ) = ( ( seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) )
14	9, 11, 13	3eqtr4g 2681	. 2 |- ( ( ph /\ M e. NN ) -> ( Q ` M ) = ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) ) )
15		0nnn 11052	. . . . . 6 |- -. 0 e. NN
16		disjsn 4246	. . . . . 6 |- ( ( NN i^i { 0 } ) = (/) <-> -. 0 e. NN )
17	15, 16	mpbir 221	. . . . 5 |- ( NN i^i { 0 } ) = (/)
18		fnresi 6008	. . . . . 6 |- ( _I |` NN ) Fn NN
19		c0ex 10034	. . . . . . 7 |- 0 e. _V
20		snex 4908	. . . . . . 7 |- { <. 0 , P >. } e. _V
21	19, 20	fnsn 5946	. . . . . 6 |- { <. 0 , { <. 0 , P >. } >. } Fn { 0 }
22		fvun1 6269	. . . . . 6 |- ( ( ( _I |` NN ) Fn NN /\ { <. 0 , { <. 0 , P >. } >. } Fn { 0 } /\ ( ( NN i^i { 0 } ) = (/) /\ M e. NN ) ) -> ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) = ( ( _I |` NN ) ` M ) )
23	18, 21, 22	mp3an12 1414	. . . . 5 |- ( ( ( NN i^i { 0 } ) = (/) /\ M e. NN ) -> ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) = ( ( _I |` NN ) ` M ) )
24	17, 2, 23	sylancr 695	. . . 4 |- ( ( ph /\ M e. NN ) -> ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) = ( ( _I |` NN ) ` M ) )
25		fvresi 6439	. . . . 5 |- ( M e. NN -> ( ( _I |` NN ) ` M ) = M )
26	25	adantl 482	. . . 4 |- ( ( ph /\ M e. NN ) -> ( ( _I |` NN ) ` M ) = M )
27	24, 26	eqtrd 2656	. . 3 |- ( ( ph /\ M e. NN ) -> ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) = M )
28	27	oveq2d 6666	. 2 |- ( ( ph /\ M e. NN ) -> ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ( ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ` M ) ) = ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) M ) )
29		fvexd 6203	. . 3 |- ( ph -> ( Q ` ( M - 1 ) ) e. _V )
30		simpr 477	. . . . . . . . 9 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> m = M )
31	30	oveq1d 6665	. . . . . . . 8 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( m - 1 ) = ( M - 1 ) )
32	31	oveq1d 6665	. . . . . . 7 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( ( m - 1 ) / N ) = ( ( M - 1 ) / N ) )
33	30	oveq1d 6665	. . . . . . 7 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( m / N ) = ( M / N ) )
34	32, 33	oveq12d 6668	. . . . . 6 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( ( ( m - 1 ) / N ) [,] ( m / N ) ) = ( ( ( M - 1 ) / N ) [,] ( M / N ) ) )
35		cvmliftlem5.3	. . . . . 6 |- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )
36	34, 35	syl6eqr 2674	. . . . 5 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( ( ( m - 1 ) / N ) [,] ( m / N ) ) = W )
37	30	fveq2d 6195	. . . . . . . . . 10 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( T ` m ) = ( T ` M ) )
38	37	fveq2d 6195	. . . . . . . . 9 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( 2nd ` ( T ` m ) ) = ( 2nd ` ( T ` M ) ) )
39		simpl 473	. . . . . . . . . . 11 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> x = ( Q ` ( M - 1 ) ) )
40	39, 32	fveq12d 6197	. . . . . . . . . 10 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( x ` ( ( m - 1 ) / N ) ) = ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) )
41	40	eleq1d 2686	. . . . . . . . 9 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( ( x ` ( ( m - 1 ) / N ) ) e. b <-> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) )
42	38, 41	riotaeqbidv 6614	. . . . . . . 8 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) = ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) )
43	42	reseq2d 5396	. . . . . . 7 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) = ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) )
44	43	cnveqd 5298	. . . . . 6 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) = `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) )
45	44	fveq1d 6193	. . . . 5 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) = ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) )
46	36, 45	mpteq12dv 4733	. . . 4 |- ( ( x = ( Q ` ( M - 1 ) ) /\ m = M ) -> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
47		eqid 2622	. . . 4 |- ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) = ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
48		ovex 6678	. . . . . 6 |- ( ( ( M - 1 ) / N ) [,] ( M / N ) ) e. _V
49	35, 48	eqeltri 2697	. . . . 5 |- W e. _V
50	49	mptex 6486	. . . 4 |- ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) e. _V
51	46, 47, 50	ovmpt2a 6791	. . 3 |- ( ( ( Q ` ( M - 1 ) ) e. _V /\ M e. NN ) -> ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
52	29, 51	sylan 488	. 2 |- ( ( ph /\ M e. NN ) -> ( ( Q ` ( M - 1 ) ) ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
53	14, 28, 52	3eqtrd 2660	1 |- ( ( ph /\ M e. NN ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   seqcseq 12801   ↾_t crest 16081   topGenctg 16098   Cn ccn 21028   Homeochmeo 21556   IIcii 22678   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802

This theorem is referenced by:  cvmliftlem6  31272  cvmliftlem8  31274  cvmliftlem9  31275