Theorem refsum2cnlem1 27575

Description: This is the core Lemma for refsum2cn 27576: 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 4258	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2537	. . . . . . . 8 |- F/_ k A
5		nfcv 2540	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5694	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2540	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5694	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2540	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5694	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5694	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		ax-1cn 9004	. . . . . 6 |- 1 e. CC
15	14	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
16		2cn 10026	. . . . . 6 |- 2 e. CC
17	16	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
18		1ex 9042	. . . . . . . . . . 11 |- 1 e. _V
19	18	prid1 3872	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
20		eqid 2404	. . . . . . . . . . . 12 |- 1 = 1
21		iftrue 3705	. . . . . . . . . . . 12 |- ( 1 = 1 -> if ( 1 = 1 , F , G ) = F )
22	20, 21	ax-mp 8	. . . . . . . . . . 11 |- if ( 1 = 1 , F , G ) = F
23		refsum2cnlem1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
24	22, 23	syl5eqel 2488	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
25		eqeq1 2410	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
26	25	ifbid 3717	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
27	26, 2	fvmptg 5763	. . . . . . . . . 10 |- ( ( 1 e. { 1 , 2 } /\ if ( 1 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
28	19, 24, 27	sylancr 645	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
29	28, 22	syl6eq 2452	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
30	29	adantr 452	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
31	30	fveq1d 5689	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
32		eqid 2404	. . . . . . . . . . 11 |- U. J = U. J
33		eqid 2404	. . . . . . . . . . 11 |- U. K = U. K
34	32, 33	cnf 17264	. . . . . . . . . 10 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
35	23, 34	syl 16	. . . . . . . . 9 |- ( ph -> F : U. J --> U. K )
36		refsum2cnlem1.7	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` X ) )
37		toponuni 16947	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
38	36, 37	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
39	38	eqcomd 2409	. . . . . . . . . 10 |- ( ph -> U. J = X )
40		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
41	40	unieqi 3985	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
42		uniretop 18749	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
43	41, 42	eqtr4i 2427	. . . . . . . . . . 11 |- U. K = RR
44	43	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
45	39, 44	feq23d 5547	. . . . . . . . 9 |- ( ph -> ( F : U. J --> U. K <-> F : X --> RR ) )
46	35, 45	mpbid 202	. . . . . . . 8 |- ( ph -> F : X --> RR )
47	46	anim1i 552	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F : X --> RR /\ x e. X ) )
48		ffvelrn 5827	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
49		recn 9036	. . . . . . 7 |- ( ( F ` x ) e. RR -> ( F ` x ) e. CC )
50	47, 48, 49	3syl 19	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
51	31, 50	eqeltrd 2478	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
52	16	elexi 2925	. . . . . . . . . . 11 |- 2 e. _V
53	52	prid2 3873	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
54		1ne2 10143	. . . . . . . . . . . . . 14 |- 1 =/= 2
55	54	necomi 2649	. . . . . . . . . . . . 13 |- 2 =/= 1
56		df-ne 2569	. . . . . . . . . . . . 13 |- ( 2 =/= 1 <-> -. 2 = 1 )
57	55, 56	mpbi 200	. . . . . . . . . . . 12 |- -. 2 = 1
58		iffalse 3706	. . . . . . . . . . . 12 |- ( -. 2 = 1 -> if ( 2 = 1 , F , G ) = G )
59	57, 58	ax-mp 8	. . . . . . . . . . 11 |- if ( 2 = 1 , F , G ) = G
60		refsum2cnlem1.9	. . . . . . . . . . 11 |- ( ph -> G e. ( J Cn K ) )
61	59, 60	syl5eqel 2488	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
62		eqeq1 2410	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
63	62	ifbid 3717	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
64	63, 2	fvmptg 5763	. . . . . . . . . 10 |- ( ( 2 e. { 1 , 2 } /\ if ( 2 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
65	53, 61, 64	sylancr 645	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
66	65, 59	syl6eq 2452	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
67	66	adantr 452	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
68	67	fveq1d 5689	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
69	32, 33	cnf 17264	. . . . . . . . . 10 |- ( G e. ( J Cn K ) -> G : U. J --> U. K )
70	60, 69	syl 16	. . . . . . . . 9 |- ( ph -> G : U. J --> U. K )
71	39, 44	feq23d 5547	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
72	70, 71	mpbid 202	. . . . . . . 8 |- ( ph -> G : X --> RR )
73	72	anim1i 552	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
74		ffvelrn 5827	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
75		recn 9036	. . . . . . 7 |- ( ( G ` x ) e. RR -> ( G ` x ) e. CC )
76	73, 74, 75	3syl 19	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
77	68, 76	eqeltrd 2478	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
78	54	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
79		fveq2 5687	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
80	79	fveq1d 5689	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
81	80	adantl 453	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
82		fveq2 5687	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
83	82	fveq1d 5689	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
84	83	adantl 453	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
85	9, 13, 15, 17, 51, 77, 78, 81, 84	sumpair 27573	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
86	31, 68	oveq12d 6058	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
87	85, 86	eqtrd 2436	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
88	1, 87	mpteq2da 4254	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
89		prfi 7340	. . . 4 |- { 1 , 2 } e. Fin
90	89	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
91		eqid 2404	. . . . . . . . . 10 |- X = X
92	91	ax-gen 1552	. . . . . . . . 9 |- A. x X = X
93		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
94		nfcv 2540	. . . . . . . . . . . 12 |- F/_ x k
95	93, 94	nffv 5694	. . . . . . . . . . 11 |- F/_ x ( A ` k )
96		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
97	95, 96	nfeq 2547	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
98		fveq1 5686	. . . . . . . . . . 11 |- ( ( A ` k ) = F -> ( ( A ` k ) ` x ) = ( F ` x ) )
99	98	a1d 23	. . . . . . . . . 10 |- ( ( A ` k ) = F -> ( x e. X -> ( ( A ` k ) ` x ) = ( F ` x ) ) )
100	97, 99	ralrimi 2747	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
101		mpteq12f 4245	. . . . . . . . 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 ) ) )
102	92, 100, 101	sylancr 645	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
103	102	adantl 453	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
104		retopon 18750	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
105	40, 104	eqeltri 2474	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
106	105	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
107		cnf2 17267	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
108	36, 106, 23, 107	syl3anc 1184	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
109		ffn 5550	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
110	108, 109	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
111	96	dffn5f 5740	. . . . . . . . 9 |- ( F Fn X <-> F = ( x e. X |-> ( F ` x ) ) )
112	110, 111	sylib 189	. . . . . . . 8 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
113	112	adantr 452	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
114	103, 113	eqtr4d 2439	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
115	23	adantr 452	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
116	114, 115	eqeltrd 2478	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
117	116	adantlr 696	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
118		refsum2cnlem1.3	. . . . . . . . . . 11 |- F/_ x G
119	95, 118	nfeq 2547	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
120		fveq1 5686	. . . . . . . . . . 11 |- ( ( A ` k ) = G -> ( ( A ` k ) ` x ) = ( G ` x ) )
121	120	a1d 23	. . . . . . . . . 10 |- ( ( A ` k ) = G -> ( x e. X -> ( ( A ` k ) ` x ) = ( G ` x ) ) )
122	119, 121	ralrimi 2747	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
123		mpteq12f 4245	. . . . . . . . 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 ) ) )
124	92, 122, 123	sylancr 645	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
125	124	adantl 453	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
126		cnf2 17267	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
127	36, 106, 60, 126	syl3anc 1184	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
128		ffn 5550	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
129	127, 128	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
130	118	dffn5f 5740	. . . . . . . . 9 |- ( G Fn X <-> G = ( x e. X |-> ( G ` x ) ) )
131	129, 130	sylib 189	. . . . . . . 8 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
132	131	adantr 452	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
133	125, 132	eqtr4d 2439	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
134	60	adantr 452	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
135	133, 134	eqeltrd 2478	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
136	135	adantlr 696	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
137		simpr 448	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
138		ifcl 3735	. . . . . . . . . 10 |- ( ( F e. ( J Cn K ) /\ G e. ( J Cn K ) ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
139	23, 60, 138	syl2anc 643	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
140	139	adantr 452	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
141	2	fvmpt2 5771	. . . . . . . 8 |- ( ( k e. { 1 , 2 } /\ if ( k = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` k ) = if ( k = 1 , F , G ) )
142	137, 140, 141	syl2anc 643	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
143		iftrue 3705	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
144	142, 143	sylan9eq 2456	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( A ` k ) = F )
145	144	orcd 382	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
146	142	adantr 452	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
147		neeq2 2576	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
148	54, 147	mpbiri 225	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
149	148	necomd 2650	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
150	149	neneqd 2583	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
151	150	adantl 453	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
152		iffalse 3706	. . . . . . . 8 |- ( -. k = 1 -> if ( k = 1 , F , G ) = G )
153	151, 152	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
154	146, 153	eqtrd 2436	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = G )
155	154	olcd 383	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
156		elpri 3794	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
157	156	adantl 453	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
158	145, 155, 157	mpjaodan 762	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
159	117, 136, 158	mpjaodan 762	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
160	1, 40, 36, 90, 159	refsumcn 27568	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
161	88, 160	eqeltrrd 2479	1 |- ( ph -> ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) e. ( J Cn K ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 358   /\ wa 359   A.wal 1546   F/wnf 1550   = wceq 1649   e. wcel 1721   F/_wnfc 2527   =/= wne 2567   A.wral 2666   ifcif 3699   {cpr 3775   U.cuni 3975   e. cmpt 4226   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   1c1 8947   + caddc 8949   2c2 10005   (,)cioo 10872   sum_csu 12434   topGenctg 13620  TopOnctopon 16914   Cn ccn 17242

This theorem is referenced by:  refsum2cn  27576

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305