Theorem refsum2cnlem1 29897

Description: This is the core Lemma for refsum2cn 29898: 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 4479	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2611	. . . . . . . 8 |- F/_ k A
5		nfcv 2613	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5796	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2613	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5796	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2613	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5796	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5796	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		ax-1cn 9441	. . . . . 6 |- 1 e. CC
15	14	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
16		2cnd 10495	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
17		1ex 9482	. . . . . . . . . . 11 |- 1 e. _V
18	17	prid1 4081	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
19		eqid 2451	. . . . . . . . . . . 12 |- 1 = 1
20	19	iftruei 3896	. . . . . . . . . . 11 |- if ( 1 = 1 , F , G ) = F
21		refsum2cnlem1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
22	20, 21	syl5eqel 2543	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
23		eqeq1 2455	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
24	23	ifbid 3909	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
25	24, 2	fvmptg 5871	. . . . . . . . . 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 2508	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5791	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2451	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2451	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 18966	. . . . . . . . . 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 18648	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2459	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4198	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 20457	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2483	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5652	. . . . . . . . 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 5940	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 9473	. . . . . . 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 2539	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10494	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4082	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52		1ne2 10635	. . . . . . . . . . . . . 14 |- 1 =/= 2
53	52	necomi 2718	. . . . . . . . . . . . 13 |- 2 =/= 1
54		df-ne 2646	. . . . . . . . . . . . 13 |- ( 2 =/= 1 <-> -. 2 = 1 )
55	53, 54	mpbi 208	. . . . . . . . . . . 12 |- -. 2 = 1
56	55	iffalsei 3898	. . . . . . . . . . 11 |- if ( 2 = 1 , F , G ) = G
57		refsum2cnlem1.9	. . . . . . . . . . 11 |- ( ph -> G e. ( J Cn K ) )
58	56, 57	syl5eqel 2543	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
59		eqeq1 2455	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
60	59	ifbid 3909	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
61	60, 2	fvmptg 5871	. . . . . . . . . 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 2508	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
64	63	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
65	64	fveq1d 5791	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
66	30, 31	cnf 18966	. . . . . . . . . 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 5652	. . . . . . . . 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 5940	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
72		recn 9473	. . . . . . 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 2539	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
75	52	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
76		fveq2 5789	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
77	76	fveq1d 5791	. . . . . 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 5789	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
80	79	fveq1d 5791	. . . . . 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 29895	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
83	29, 65	oveq12d 6208	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
84	82, 83	eqtrd 2492	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
85	1, 84	mpteq2da 4475	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
86		prfi 7687	. . . 4 |- { 1 , 2 } e. Fin
87	86	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
88		eqid 2451	. . . . . . . . . 10 |- X = X
89	88	ax-gen 1592	. . . . . . . . 9 |- A. x X = X
90		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
91		nfcv 2613	. . . . . . . . . . . 12 |- F/_ x k
92	90, 91	nffv 5796	. . . . . . . . . . 11 |- F/_ x ( A ` k )
93		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
94	92, 93	nfeq 2623	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
95		fveq1 5788	. . . . . . . . . . 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 2815	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
98		mpteq12f 4466	. . . . . . . . 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 20458	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
102	38, 101	eqeltri 2535	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
103	102	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
104		cnf2 18969	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
105	34, 103, 21, 104	syl3anc 1219	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
106		ffn 5657	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
107	105, 106	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
108	93	dffn5f 5845	. . . . . . . . 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 2495	. . . . . 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 2539	. . . . 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 2623	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
117		fveq1 5788	. . . . . . . . . . 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 2815	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
120		mpteq12f 4466	. . . . . . . . 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 18969	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
124	34, 103, 57, 123	syl3anc 1219	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
125		ffn 5657	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
126	124, 125	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
127	115	dffn5f 5845	. . . . . . . . 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 2495	. . . . . 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 2539	. . . . 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 3929	. . . . . . . . . 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 5880	. . . . . . . 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 3895	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
141	139, 140	sylan9eq 2512	. . . . . 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 2731	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
145	52, 144	mpbiri 233	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
146	145	necomd 2719	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
147	146	neneqd 2651	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
148	147	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
149		iffalse 3897	. . . . . . . 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 2492	. . . . . 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 3995	. . . . . 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 29890	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
158	85, 157	eqeltrrd 2540	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 1368   = wceq 1370   F/wnf 1590   e. wcel 1758   F/_wnfc 2599   =/= wne 2644   A.wral 2795   ifcif 3889   {cpr 3977   U.cuni 4189   |-> cmpt 4448   ran crn 4939   Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   Fincfn 7410   CCcc 9381   RRcr 9382   1c1 9384   + caddc 9386   2c2 10472   (,)cioo 11401   sum_csu 13265   topGenctg 14478  TopOnctopon 18615   Cn ccn 18944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cn 18947  df-cnp 18948  df-tx 19251  df-hmeo 19444  df-xms 20011  df-ms 20012  df-tms 20013

This theorem is referenced by:  refsum2cn  29898