Theorem refsum2cnlem1 30809

Description: This is the core Lemma for refsum2cn 30810: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
refsum2cnlem1.1	|- F/_ x A
refsum2cnlem1.2	|- F/_ x F
refsum2cnlem1.3	|- F/_ x G
refsum2cnlem1.4	|- F/ x ph
refsum2cnlem1.5	|- A = ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
refsum2cnlem1.6	|- K = ( topGen ` ran (,) )
refsum2cnlem1.7	|- ( ph -> J e. (TopOn ` X ) )
refsum2cnlem1.8	|- ( ph -> F e. ( J Cn K ) )
refsum2cnlem1.9	|- ( ph -> G e. ( J Cn K ) )

Assertion

Ref	Expression
refsum2cnlem1	|- ( ph -> ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) e. ( J Cn K ) )

Distinct variable groups:   x, k, J   k, F   k, G   k, K, x   k, X, x   ph, k

Allowed substitution hints:   ph( x)   A( x, k)   F( x)   G( x)

Proof of Theorem refsum2cnlem1

Step	Hyp	Ref	Expression
1		refsum2cnlem1.4	. . 3 |- F/ x ph
2		refsum2cnlem1.5	. . . . . . . . 9 |- A = ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
3		nfmpt1 4529	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2620	. . . . . . . 8 |- F/_ k A
5		nfcv 2622	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5864	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2622	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5864	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2622	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5864	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5864	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		ax-1cn 9539	. . . . . 6 |- 1 e. CC
15	14	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
16		2cnd 10597	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
17		1ex 9580	. . . . . . . . . . 11 |- 1 e. _V
18	17	prid1 4128	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
19		eqid 2460	. . . . . . . . . . . 12 |- 1 = 1
20	19	iftruei 3939	. . . . . . . . . . 11 |- if ( 1 = 1 , F , G ) = F
21		refsum2cnlem1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
22	20, 21	syl5eqel 2552	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
23		eqeq1 2464	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
24	23	ifbid 3954	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
25	24, 2	fvmptg 5939	. . . . . . . . . 10 |- ( ( 1 e. { 1 , 2 } /\ if ( 1 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
26	18, 22, 25	sylancr 663	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
27	26, 20	syl6eq 2517	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5859	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2460	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2460	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 19506	. . . . . . . . . 10 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
33	21, 32	syl 16	. . . . . . . . 9 |- ( ph -> F : U. J --> U. K )
34		refsum2cnlem1.7	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` X ) )
35		toponuni 19188	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2468	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4247	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 20997	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2492	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5717	. . . . . . . . 9 |- ( ph -> ( F : U. J --> U. K <-> F : X --> RR ) )
44	33, 43	mpbid 210	. . . . . . . 8 |- ( ph -> F : X --> RR )
45	44	anim1i 568	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F : X --> RR /\ x e. X ) )
46		ffvelrn 6010	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 9571	. . . . . . 7 |- ( ( F ` x ) e. RR -> ( F ` x ) e. CC )
48	45, 46, 47	3syl 20	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
49	29, 48	eqeltrd 2548	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10596	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4129	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52		1ne2 10737	. . . . . . . . . . . . . 14 |- 1 =/= 2
53	52	necomi 2730	. . . . . . . . . . . . 13 |- 2 =/= 1
54		df-ne 2657	. . . . . . . . . . . . 13 |- ( 2 =/= 1 <-> -. 2 = 1 )
55	53, 54	mpbi 208	. . . . . . . . . . . 12 |- -. 2 = 1
56	55	iffalsei 3942	. . . . . . . . . . 11 |- if ( 2 = 1 , F , G ) = G
57		refsum2cnlem1.9	. . . . . . . . . . 11 |- ( ph -> G e. ( J Cn K ) )
58	56, 57	syl5eqel 2552	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
59		eqeq1 2464	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
60	59	ifbid 3954	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
61	60, 2	fvmptg 5939	. . . . . . . . . 10 |- ( ( 2 e. { 1 , 2 } /\ if ( 2 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
62	51, 58, 61	sylancr 663	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
63	62, 56	syl6eq 2517	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
64	63	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
65	64	fveq1d 5859	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
66	30, 31	cnf 19506	. . . . . . . . . 10 |- ( G e. ( J Cn K ) -> G : U. J --> U. K )
67	57, 66	syl 16	. . . . . . . . 9 |- ( ph -> G : U. J --> U. K )
68	37, 42	feq23d 5717	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
69	67, 68	mpbid 210	. . . . . . . 8 |- ( ph -> G : X --> RR )
70	69	anim1i 568	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
71		ffvelrn 6010	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
72		recn 9571	. . . . . . 7 |- ( ( G ` x ) e. RR -> ( G ` x ) e. CC )
73	70, 71, 72	3syl 20	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
74	65, 73	eqeltrd 2548	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
75	52	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
76		fveq2 5857	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
77	76	fveq1d 5859	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
78	77	adantl 466	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
79		fveq2 5857	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
80	79	fveq1d 5859	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
81	80	adantl 466	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
82	9, 13, 15, 16, 49, 74, 75, 78, 81	sumpair 30807	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
83	29, 65	oveq12d 6293	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
84	82, 83	eqtrd 2501	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
85	1, 84	mpteq2da 4525	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
86		prfi 7784	. . . 4 |- { 1 , 2 } e. Fin
87	86	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
88		eqid 2460	. . . . . . . . . 10 |- X = X
89	88	ax-gen 1596	. . . . . . . . 9 |- A. x X = X
90		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
91		nfcv 2622	. . . . . . . . . . . 12 |- F/_ x k
92	90, 91	nffv 5864	. . . . . . . . . . 11 |- F/_ x ( A ` k )
93		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
94	92, 93	nfeq 2633	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
95		fveq1 5856	. . . . . . . . . . 11 |- ( ( A ` k ) = F -> ( ( A ` k ) ` x ) = ( F ` x ) )
96	95	a1d 25	. . . . . . . . . 10 |- ( ( A ` k ) = F -> ( x e. X -> ( ( A ` k ) ` x ) = ( F ` x ) ) )
97	94, 96	ralrimi 2857	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
98		mpteq12f 4516	. . . . . . . . 9 |- ( ( A. x X = X /\ A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
99	89, 97, 98	sylancr 663	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
100	99	adantl 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
101		retopon 20998	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
102	38, 101	eqeltri 2544	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
103	102	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
104		cnf2 19509	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
105	34, 103, 21, 104	syl3anc 1223	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
106		ffn 5722	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
107	105, 106	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
108	93	dffn5f 5913	. . . . . . . . 9 |- ( F Fn X <-> F = ( x e. X |-> ( F ` x ) ) )
109	107, 108	sylib 196	. . . . . . . 8 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
110	109	adantr 465	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
111	100, 110	eqtr4d 2504	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
112	21	adantr 465	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
113	111, 112	eqeltrd 2548	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
114	113	adantlr 714	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
115		refsum2cnlem1.3	. . . . . . . . . . 11 |- F/_ x G
116	92, 115	nfeq 2633	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
117		fveq1 5856	. . . . . . . . . . 11 |- ( ( A ` k ) = G -> ( ( A ` k ) ` x ) = ( G ` x ) )
118	117	a1d 25	. . . . . . . . . 10 |- ( ( A ` k ) = G -> ( x e. X -> ( ( A ` k ) ` x ) = ( G ` x ) ) )
119	116, 118	ralrimi 2857	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
120		mpteq12f 4516	. . . . . . . . 9 |- ( ( A. x X = X /\ A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
121	89, 119, 120	sylancr 663	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
122	121	adantl 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
123		cnf2 19509	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
124	34, 103, 57, 123	syl3anc 1223	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
125		ffn 5722	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
126	124, 125	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
127	115	dffn5f 5913	. . . . . . . . 9 |- ( G Fn X <-> G = ( x e. X |-> ( G ` x ) ) )
128	126, 127	sylib 196	. . . . . . . 8 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
129	128	adantr 465	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
130	122, 129	eqtr4d 2504	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
131	57	adantr 465	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
132	130, 131	eqeltrd 2548	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
133	132	adantlr 714	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
134		simpr 461	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
135		ifcl 3974	. . . . . . . . . 10 |- ( ( F e. ( J Cn K ) /\ G e. ( J Cn K ) ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
136	21, 57, 135	syl2anc 661	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
137	136	adantr 465	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
138	2	fvmpt2 5948	. . . . . . . 8 |- ( ( k e. { 1 , 2 } /\ if ( k = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` k ) = if ( k = 1 , F , G ) )
139	134, 137, 138	syl2anc 661	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
140		iftrue 3938	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
141	139, 140	sylan9eq 2521	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( A ` k ) = F )
142	141	orcd 392	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
143	139	adantr 465	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
144		neeq2 2743	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
145	52, 144	mpbiri 233	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
146	145	necomd 2731	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
147	146	neneqd 2662	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
148	147	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
149		iffalse 3941	. . . . . . . 8 |- ( -. k = 1 -> if ( k = 1 , F , G ) = G )
150	148, 149	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
151	143, 150	eqtrd 2501	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = G )
152	151	olcd 393	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
153		elpri 4040	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
154	153	adantl 466	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
155	142, 152, 154	mpjaodan 784	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
156	114, 133, 155	mpjaodan 784	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
157	1, 38, 34, 87, 156	refsumcn 30802	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
158	85, 157	eqeltrrd 2549	1 |- ( ph -> ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) e. ( J Cn K ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   A.wal 1372   = wceq 1374   F/wnf 1594   e. wcel 1762   F/_wnfc 2608   =/= wne 2655   A.wral 2807   ifcif 3932   {cpr 4022   U.cuni 4238   |-> cmpt 4498   ran crn 4993   Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   RRcr 9480   1c1 9482   + caddc 9484   2c2 10574   (,)cioo 11518   sum_csu 13457   topGenctg 14682  TopOnctopon 19155   Cn ccn 19484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cn 19487  df-cnp 19488  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553

This theorem is referenced by:  refsum2cn  30810