Theorem refsum2cnlem1 29604

Description: This is the core Lemma for refsum2cn 29605: 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 4369	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2566	. . . . . . . 8 |- F/_ k A
5		nfcv 2569	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5686	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2569	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5686	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2569	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5686	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5686	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		ax-1cn 9328	. . . . . 6 |- 1 e. CC
15	14	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
16		2cnd 10382	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
17		1ex 9369	. . . . . . . . . . 11 |- 1 e. _V
18	17	prid1 3971	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
19		eqid 2433	. . . . . . . . . . . 12 |- 1 = 1
20	19	iftruei 3786	. . . . . . . . . . 11 |- if ( 1 = 1 , F , G ) = F
21		refsum2cnlem1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
22	20, 21	syl5eqel 2517	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
23		eqeq1 2439	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
24	23	ifbid 3799	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
25	24, 2	fvmptg 5760	. . . . . . . . . 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 656	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
27	26, 20	syl6eq 2481	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 462	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5681	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2433	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2433	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 18692	. . . . . . . . . 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 18374	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2438	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4088	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 20183	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2456	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5542	. . . . . . . . 9 |- ( ph -> ( F : U. J --> U. K <-> F : X --> RR ) )
44	33, 43	mpbid 210	. . . . . . . 8 |- ( ph -> F : X --> RR )
45	44	anim1i 563	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F : X --> RR /\ x e. X ) )
46		ffvelrn 5829	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 9360	. . . . . . 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 2507	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10381	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 3972	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52		1ne2 10522	. . . . . . . . . . . . . 14 |- 1 =/= 2
53	52	necomi 2684	. . . . . . . . . . . . 13 |- 2 =/= 1
54		df-ne 2598	. . . . . . . . . . . . 13 |- ( 2 =/= 1 <-> -. 2 = 1 )
55	53, 54	mpbi 208	. . . . . . . . . . . 12 |- -. 2 = 1
56	55	iffalsei 3788	. . . . . . . . . . 11 |- if ( 2 = 1 , F , G ) = G
57		refsum2cnlem1.9	. . . . . . . . . . 11 |- ( ph -> G e. ( J Cn K ) )
58	56, 57	syl5eqel 2517	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
59		eqeq1 2439	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
60	59	ifbid 3799	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
61	60, 2	fvmptg 5760	. . . . . . . . . 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 656	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
63	62, 56	syl6eq 2481	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
64	63	adantr 462	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
65	64	fveq1d 5681	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
66	30, 31	cnf 18692	. . . . . . . . . 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 5542	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
69	67, 68	mpbid 210	. . . . . . . 8 |- ( ph -> G : X --> RR )
70	69	anim1i 563	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
71		ffvelrn 5829	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
72		recn 9360	. . . . . . 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 2507	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
75	52	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
76		fveq2 5679	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
77	76	fveq1d 5681	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
78	77	adantl 463	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
79		fveq2 5679	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
80	79	fveq1d 5681	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
81	80	adantl 463	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
82	9, 13, 15, 16, 49, 74, 75, 78, 81	sumpair 29602	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
83	29, 65	oveq12d 6098	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
84	82, 83	eqtrd 2465	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
85	1, 84	mpteq2da 4365	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
86		prfi 7574	. . . 4 |- { 1 , 2 } e. Fin
87	86	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
88		eqid 2433	. . . . . . . . . 10 |- X = X
89	88	ax-gen 1594	. . . . . . . . 9 |- A. x X = X
90		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
91		nfcv 2569	. . . . . . . . . . . 12 |- F/_ x k
92	90, 91	nffv 5686	. . . . . . . . . . 11 |- F/_ x ( A ` k )
93		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
94	92, 93	nfeq 2576	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
95		fveq1 5678	. . . . . . . . . . 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 2787	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
98		mpteq12f 4356	. . . . . . . . 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 656	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
100	99	adantl 463	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
101		retopon 20184	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
102	38, 101	eqeltri 2503	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
103	102	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
104		cnf2 18695	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
105	34, 103, 21, 104	syl3anc 1211	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
106		ffn 5547	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
107	105, 106	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
108	93	dffn5f 5734	. . . . . . . . 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 462	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
111	100, 110	eqtr4d 2468	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
112	21	adantr 462	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
113	111, 112	eqeltrd 2507	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
114	113	adantlr 707	. . . 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 2576	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
117		fveq1 5678	. . . . . . . . . . 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 2787	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
120		mpteq12f 4356	. . . . . . . . 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 656	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
122	121	adantl 463	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
123		cnf2 18695	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
124	34, 103, 57, 123	syl3anc 1211	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
125		ffn 5547	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
126	124, 125	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
127	115	dffn5f 5734	. . . . . . . . 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 462	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
130	122, 129	eqtr4d 2468	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
131	57	adantr 462	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
132	130, 131	eqeltrd 2507	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
133	132	adantlr 707	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
134		simpr 458	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
135		ifcl 3819	. . . . . . . . . 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 654	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
137	136	adantr 462	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
138	2	fvmpt2 5769	. . . . . . . 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 654	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
140		iftrue 3785	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
141	139, 140	sylan9eq 2485	. . . . . 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 462	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
144		neeq2 2607	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
145	52, 144	mpbiri 233	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
146	145	necomd 2685	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
147	146	neneqd 2614	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
148	147	adantl 463	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
149		iffalse 3787	. . . . . . . 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 2465	. . . . . 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 3885	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
154	153	adantl 463	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
155	142, 152, 154	mpjaodan 777	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
156	114, 133, 155	mpjaodan 777	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
157	1, 38, 34, 87, 156	refsumcn 29597	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
158	85, 157	eqeltrrd 2508	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 1360   = wceq 1362   F/wnf 1592   e. wcel 1755   F/_wnfc 2556   =/= wne 2596   A.wral 2705   ifcif 3779   {cpr 3867   U.cuni 4079   e. cmpt 4338   ran crn 4828   Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   RRcr 9269   1c1 9271   + caddc 9273   2c2 10359   (,)cioo 11288   sum_csu 13147   topGenctg 14359  TopOnctopon 18341   Cn ccn 18670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cn 18673  df-cnp 18674  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739

This theorem is referenced by:  refsum2cn  29605