Theorem vdwlem1 15685

Description: Lemma for vdw 15698. (Contributed by Mario Carneiro, 12-Sep-2014.)

Hypotheses

Ref	Expression
vdwlem1.r	|- ( ph -> R e. Fin )
vdwlem1.k	|- ( ph -> K e. NN )
vdwlem1.w	|- ( ph -> W e. NN )
vdwlem1.f	|- ( ph -> F : ( 1 ... W ) --> R )
vdwlem1.a	|- ( ph -> A e. NN )
vdwlem1.m	|- ( ph -> M e. NN )
vdwlem1.d	|- ( ph -> D : ( 1 ... M ) --> NN )
vdwlem1.s	|- ( ph -> A. i e. ( 1 ... M ) ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) )
vdwlem1.i	|- ( ph -> I e. ( 1 ... M ) )
vdwlem1.e	|- ( ph -> ( F ` A ) = ( F ` ( A + ( D ` I ) ) ) )

Assertion

Ref	Expression
vdwlem1	|- ( ph -> ( K + 1 ) MonoAP F )

Distinct variable groups:   A, i   D, i   i, I   i, K   i, F   i, M   ph, i   R, i   i, W

Proof of Theorem vdwlem1

Dummy variables a c d m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwlem1.a	. . . 4 |- ( ph -> A e. NN )
2		vdwlem1.d	. . . . 5 |- ( ph -> D : ( 1 ... M ) --> NN )
3		vdwlem1.i	. . . . 5 |- ( ph -> I e. ( 1 ... M ) )
4	2, 3	ffvelrnd 6360	. . . 4 |- ( ph -> ( D ` I ) e. NN )
5		vdwlem1.k	. . . . . . 7 |- ( ph -> K e. NN )
6	5	nnnn0d 11351	. . . . . 6 |- ( ph -> K e. NN0 )
7		vdwapun 15678	. . . . . 6 |- ( ( K e. NN0 /\ A e. NN /\ ( D ` I ) e. NN ) -> ( A (AP ` ( K + 1 ) ) ( D ` I ) ) = ( { A } u. ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) ) )
8	6, 1, 4, 7	syl3anc 1326	. . . . 5 |- ( ph -> ( A (AP ` ( K + 1 ) ) ( D ` I ) ) = ( { A } u. ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) ) )
9	1	nnred 11035	. . . . . . . . . 10 |- ( ph -> A e. RR )
10		vdwlem1.m	. . . . . . . . . . . . . . 15 |- ( ph -> M e. NN )
11		nnuz 11723	. . . . . . . . . . . . . . 15 |- NN = ( ZZ>= ` 1 )
12	10, 11	syl6eleq 2711	. . . . . . . . . . . . . 14 |- ( ph -> M e. ( ZZ>= ` 1 ) )
13		eluzfz1 12348	. . . . . . . . . . . . . 14 |- ( M e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... M ) )
14	12, 13	syl 17	. . . . . . . . . . . . 13 |- ( ph -> 1 e. ( 1 ... M ) )
15	2, 14	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ph -> ( D ` 1 ) e. NN )
16	1, 15	nnaddcld 11067	. . . . . . . . . . 11 |- ( ph -> ( A + ( D ` 1 ) ) e. NN )
17	16	nnred 11035	. . . . . . . . . 10 |- ( ph -> ( A + ( D ` 1 ) ) e. RR )
18		vdwlem1.w	. . . . . . . . . . 11 |- ( ph -> W e. NN )
19	18	nnred 11035	. . . . . . . . . 10 |- ( ph -> W e. RR )
20	15	nnrpd 11870	. . . . . . . . . . . 12 |- ( ph -> ( D ` 1 ) e. RR+ )
21	9, 20	ltaddrpd 11905	. . . . . . . . . . 11 |- ( ph -> A < ( A + ( D ` 1 ) ) )
22	9, 17, 21	ltled 10185	. . . . . . . . . 10 |- ( ph -> A <_ ( A + ( D ` 1 ) ) )
23		vdwlem1.s	. . . . . . . . . . . . . . . 16 |- ( ph -> A. i e. ( 1 ... M ) ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) )
24	23	r19.21bi 2932	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) )
25		cnvimass 5485	. . . . . . . . . . . . . . . . 17 |- ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) C_ dom F
26		vdwlem1.f	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F : ( 1 ... W ) --> R )
27		fdm 6051	. . . . . . . . . . . . . . . . . 18 |- ( F : ( 1 ... W ) --> R -> dom F = ( 1 ... W ) )
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> dom F = ( 1 ... W ) )
29	25, 28	syl5sseq 3653	. . . . . . . . . . . . . . . 16 |- ( ph -> ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) C_ ( 1 ... W ) )
30	29	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) C_ ( 1 ... W ) )
31	24, 30	sstrd 3613	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( 1 ... W ) )
32		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . 20 |- ( K e. NN -> ( K - 1 ) e. NN0 )
33	5, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( K - 1 ) e. NN0 )
34		nn0uz 11722	. . . . . . . . . . . . . . . . . . 19 |- NN0 = ( ZZ>= ` 0 )
35	33, 34	syl6eleq 2711	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( K - 1 ) e. ( ZZ>= ` 0 ) )
36		eluzfz1 12348	. . . . . . . . . . . . . . . . . 18 |- ( ( K - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( K - 1 ) ) )
37	35, 36	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> 0 e. ( 0 ... ( K - 1 ) ) )
38	37	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 1 ... M ) ) -> 0 e. ( 0 ... ( K - 1 ) ) )
39	2	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( D ` i ) e. NN )
40	39	nncnd 11036	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( D ` i ) e. CC )
41	40	mul02d 10234	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( 0 x. ( D ` i ) ) = 0 )
42	41	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( A + ( D ` i ) ) + ( 0 x. ( D ` i ) ) ) = ( ( A + ( D ` i ) ) + 0 ) )
43	1	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. ( 1 ... M ) ) -> A e. NN )
44	43, 39	nnaddcld 11067	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( A + ( D ` i ) ) e. NN )
45	44	nncnd 11036	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( A + ( D ` i ) ) e. CC )
46	45	addid1d 10236	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( A + ( D ` i ) ) + 0 ) = ( A + ( D ` i ) ) )
47	42, 46	eqtr2d 2657	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( 0 x. ( D ` i ) ) ) )
48		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( m = 0 -> ( m x. ( D ` i ) ) = ( 0 x. ( D ` i ) ) )
49	48	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( m = 0 -> ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) = ( ( A + ( D ` i ) ) + ( 0 x. ( D ` i ) ) ) )
50	49	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( m = 0 -> ( ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) <-> ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( 0 x. ( D ` i ) ) ) ) )
51	50	rspcev 3309	. . . . . . . . . . . . . . . 16 |- ( ( 0 e. ( 0 ... ( K - 1 ) ) /\ ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( 0 x. ( D ` i ) ) ) ) -> E. m e. ( 0 ... ( K - 1 ) ) ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) )
52	38, 47, 51	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 1 ... M ) ) -> E. m e. ( 0 ... ( K - 1 ) ) ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) )
53	5	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ i e. ( 1 ... M ) ) -> K e. NN )
54	53	nnnn0d 11351	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 1 ... M ) ) -> K e. NN0 )
55		vdwapval 15677	. . . . . . . . . . . . . . . 16 |- ( ( K e. NN0 /\ ( A + ( D ` i ) ) e. NN /\ ( D ` i ) e. NN ) -> ( ( A + ( D ` i ) ) e. ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) <-> E. m e. ( 0 ... ( K - 1 ) ) ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) ) )
56	54, 44, 39, 55	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( A + ( D ` i ) ) e. ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) <-> E. m e. ( 0 ... ( K - 1 ) ) ( A + ( D ` i ) ) = ( ( A + ( D ` i ) ) + ( m x. ( D ` i ) ) ) ) )
57	52, 56	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( A + ( D ` i ) ) e. ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) )
58	31, 57	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( A + ( D ` i ) ) e. ( 1 ... W ) )
59	58	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ph -> A. i e. ( 1 ... M ) ( A + ( D ` i ) ) e. ( 1 ... W ) )
60		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( i = 1 -> ( D ` i ) = ( D ` 1 ) )
61	60	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( i = 1 -> ( A + ( D ` i ) ) = ( A + ( D ` 1 ) ) )
62	61	eleq1d 2686	. . . . . . . . . . . . 13 |- ( i = 1 -> ( ( A + ( D ` i ) ) e. ( 1 ... W ) <-> ( A + ( D ` 1 ) ) e. ( 1 ... W ) ) )
63	62	rspcv 3305	. . . . . . . . . . . 12 |- ( 1 e. ( 1 ... M ) -> ( A. i e. ( 1 ... M ) ( A + ( D ` i ) ) e. ( 1 ... W ) -> ( A + ( D ` 1 ) ) e. ( 1 ... W ) ) )
64	14, 59, 63	sylc 65	. . . . . . . . . . 11 |- ( ph -> ( A + ( D ` 1 ) ) e. ( 1 ... W ) )
65		elfzle2 12345	. . . . . . . . . . 11 |- ( ( A + ( D ` 1 ) ) e. ( 1 ... W ) -> ( A + ( D ` 1 ) ) <_ W )
66	64, 65	syl 17	. . . . . . . . . 10 |- ( ph -> ( A + ( D ` 1 ) ) <_ W )
67	9, 17, 19, 22, 66	letrd 10194	. . . . . . . . 9 |- ( ph -> A <_ W )
68	1, 11	syl6eleq 2711	. . . . . . . . . 10 |- ( ph -> A e. ( ZZ>= ` 1 ) )
69	18	nnzd 11481	. . . . . . . . . 10 |- ( ph -> W e. ZZ )
70		elfz5 12334	. . . . . . . . . 10 |- ( ( A e. ( ZZ>= ` 1 ) /\ W e. ZZ ) -> ( A e. ( 1 ... W ) <-> A <_ W ) )
71	68, 69, 70	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( A e. ( 1 ... W ) <-> A <_ W ) )
72	67, 71	mpbird 247	. . . . . . . 8 |- ( ph -> A e. ( 1 ... W ) )
73		eqidd 2623	. . . . . . . 8 |- ( ph -> ( F ` A ) = ( F ` A ) )
74		ffn 6045	. . . . . . . . 9 |- ( F : ( 1 ... W ) --> R -> F Fn ( 1 ... W ) )
75		fniniseg 6338	. . . . . . . . 9 |- ( F Fn ( 1 ... W ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> ( A e. ( 1 ... W ) /\ ( F ` A ) = ( F ` A ) ) ) )
76	26, 74, 75	3syl 18	. . . . . . . 8 |- ( ph -> ( A e. ( `' F " { ( F ` A ) } ) <-> ( A e. ( 1 ... W ) /\ ( F ` A ) = ( F ` A ) ) ) )
77	72, 73, 76	mpbir2and 957	. . . . . . 7 |- ( ph -> A e. ( `' F " { ( F ` A ) } ) )
78	77	snssd 4340	. . . . . 6 |- ( ph -> { A } C_ ( `' F " { ( F ` A ) } ) )
79		fveq2 6191	. . . . . . . . . . . 12 |- ( i = I -> ( D ` i ) = ( D ` I ) )
80	79	oveq2d 6666	. . . . . . . . . . 11 |- ( i = I -> ( A + ( D ` i ) ) = ( A + ( D ` I ) ) )
81	80, 79	oveq12d 6668	. . . . . . . . . 10 |- ( i = I -> ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) = ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) )
82	80	fveq2d 6195	. . . . . . . . . . . 12 |- ( i = I -> ( F ` ( A + ( D ` i ) ) ) = ( F ` ( A + ( D ` I ) ) ) )
83	82	sneqd 4189	. . . . . . . . . . 11 |- ( i = I -> { ( F ` ( A + ( D ` i ) ) ) } = { ( F ` ( A + ( D ` I ) ) ) } )
84	83	imaeq2d 5466	. . . . . . . . . 10 |- ( i = I -> ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) = ( `' F " { ( F ` ( A + ( D ` I ) ) ) } ) )
85	81, 84	sseq12d 3634	. . . . . . . . 9 |- ( i = I -> ( ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) <-> ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) C_ ( `' F " { ( F ` ( A + ( D ` I ) ) ) } ) ) )
86	85	rspcv 3305	. . . . . . . 8 |- ( I e. ( 1 ... M ) -> ( A. i e. ( 1 ... M ) ( ( A + ( D ` i ) ) (AP ` K ) ( D ` i ) ) C_ ( `' F " { ( F ` ( A + ( D ` i ) ) ) } ) -> ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) C_ ( `' F " { ( F ` ( A + ( D ` I ) ) ) } ) ) )
87	3, 23, 86	sylc 65	. . . . . . 7 |- ( ph -> ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) C_ ( `' F " { ( F ` ( A + ( D ` I ) ) ) } ) )
88		vdwlem1.e	. . . . . . . . 9 |- ( ph -> ( F ` A ) = ( F ` ( A + ( D ` I ) ) ) )
89	88	sneqd 4189	. . . . . . . 8 |- ( ph -> { ( F ` A ) } = { ( F ` ( A + ( D ` I ) ) ) } )
90	89	imaeq2d 5466	. . . . . . 7 |- ( ph -> ( `' F " { ( F ` A ) } ) = ( `' F " { ( F ` ( A + ( D ` I ) ) ) } ) )
91	87, 90	sseqtr4d 3642	. . . . . 6 |- ( ph -> ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) C_ ( `' F " { ( F ` A ) } ) )
92	78, 91	unssd 3789	. . . . 5 |- ( ph -> ( { A } u. ( ( A + ( D ` I ) ) (AP ` K ) ( D ` I ) ) ) C_ ( `' F " { ( F ` A ) } ) )
93	8, 92	eqsstrd 3639	. . . 4 |- ( ph -> ( A (AP ` ( K + 1 ) ) ( D ` I ) ) C_ ( `' F " { ( F ` A ) } ) )
94		oveq1 6657	. . . . . 6 |- ( a = A -> ( a (AP ` ( K + 1 ) ) d ) = ( A (AP ` ( K + 1 ) ) d ) )
95	94	sseq1d 3632	. . . . 5 |- ( a = A -> ( ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) <-> ( A (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) ) )
96		oveq2 6658	. . . . . 6 |- ( d = ( D ` I ) -> ( A (AP ` ( K + 1 ) ) d ) = ( A (AP ` ( K + 1 ) ) ( D ` I ) ) )
97	96	sseq1d 3632	. . . . 5 |- ( d = ( D ` I ) -> ( ( A (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) <-> ( A (AP ` ( K + 1 ) ) ( D ` I ) ) C_ ( `' F " { ( F ` A ) } ) ) )
98	95, 97	rspc2ev 3324	. . . 4 |- ( ( A e. NN /\ ( D ` I ) e. NN /\ ( A (AP ` ( K + 1 ) ) ( D ` I ) ) C_ ( `' F " { ( F ` A ) } ) ) -> E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) )
99	1, 4, 93, 98	syl3anc 1326	. . 3 |- ( ph -> E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) )
100		fvex 6201	. . . 4 |- ( F ` A ) e. _V
101		sneq 4187	. . . . . . 7 |- ( c = ( F ` A ) -> { c } = { ( F ` A ) } )
102	101	imaeq2d 5466	. . . . . 6 |- ( c = ( F ` A ) -> ( `' F " { c } ) = ( `' F " { ( F ` A ) } ) )
103	102	sseq2d 3633	. . . . 5 |- ( c = ( F ` A ) -> ( ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { c } ) <-> ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) ) )
104	103	2rexbidv 3057	. . . 4 |- ( c = ( F ` A ) -> ( E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { c } ) <-> E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) ) )
105	100, 104	spcev 3300	. . 3 |- ( E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { ( F ` A ) } ) -> E. c E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { c } ) )
106	99, 105	syl 17	. 2 |- ( ph -> E. c E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { c } ) )
107		ovex 6678	. . 3 |- ( 1 ... W ) e. _V
108		peano2nn0 11333	. . . 4 |- ( K e. NN0 -> ( K + 1 ) e. NN0 )
109	6, 108	syl 17	. . 3 |- ( ph -> ( K + 1 ) e. NN0 )
110	107, 109, 26	vdwmc 15682	. 2 |- ( ph -> ( ( K + 1 ) MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` ( K + 1 ) ) d ) C_ ( `' F " { c } ) ) )
111	106, 110	mpbird 247	1 |- ( ph -> ( K + 1 ) MonoAP F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  APcvdwa 15669   MonoAP cvdwm 15670

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-1st 7168  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-rp 11833  df-fz 12327  df-vdwap 15672  df-vdwmc 15673

This theorem is referenced by:  vdwlem6  15690