Theorem refsum2cnlem1 31359

Description: This is the core Lemma for refsum2cn 31360: 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 4522	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2601	. . . . . . . 8 |- F/_ k A
5		nfcv 2603	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5859	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2603	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5859	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2603	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5859	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5859	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		1cnd 9610	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
15		2cnd 10609	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
16		1ex 9589	. . . . . . . . . . 11 |- 1 e. _V
17	16	prid1 4119	. . . . . . . . . 10 |- 1 e. { 1 , 2 }
18		refsum2cnlem1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
19		refsum2cnlem1.9	. . . . . . . . . . 11 |- ( ph -> G e. ( J Cn K ) )
20	18, 19	ifcld 3965	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
21		eqeq1 2445	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
22	21	ifbid 3944	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
23	22, 2	fvmptg 5935	. . . . . . . . . 10 |- ( ( 1 e. { 1 , 2 } /\ if ( 1 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
24	17, 20, 23	sylancr 663	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
25		eqid 2441	. . . . . . . . . 10 |- 1 = 1
26	25	iftruei 3929	. . . . . . . . 9 |- if ( 1 = 1 , F , G ) = F
27	24, 26	syl6eq 2498	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5854	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2441	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2441	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 19613	. . . . . . . . . 10 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
33	18, 32	syl 16	. . . . . . . . 9 |- ( ph -> F : U. J --> U. K )
34		refsum2cnlem1.7	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` X ) )
35		toponuni 19295	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2449	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4239	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 21135	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2473	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5712	. . . . . . . . 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 9580	. . . . . . 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 2529	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10608	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4120	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52	18, 19	ifcld 3965	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
53		eqeq1 2445	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
54	53	ifbid 3944	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
55	54, 2	fvmptg 5935	. . . . . . . . . 10 |- ( ( 2 e. { 1 , 2 } /\ if ( 2 = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
56	51, 52, 55	sylancr 663	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
57		1ne2 10749	. . . . . . . . . . 11 |- 1 =/= 2
58	57	nesymi 2714	. . . . . . . . . 10 |- -. 2 = 1
59	58	iffalsei 3932	. . . . . . . . 9 |- if ( 2 = 1 , F , G ) = G
60	56, 59	syl6eq 2498	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
61	60	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
62	61	fveq1d 5854	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
63	30, 31	cnf 19613	. . . . . . . . . 10 |- ( G e. ( J Cn K ) -> G : U. J --> U. K )
64	19, 63	syl 16	. . . . . . . . 9 |- ( ph -> G : U. J --> U. K )
65	37, 42	feq23d 5712	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
66	64, 65	mpbid 210	. . . . . . . 8 |- ( ph -> G : X --> RR )
67	66	anim1i 568	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
68		ffvelrn 6010	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
69		recn 9580	. . . . . . 7 |- ( ( G ` x ) e. RR -> ( G ` x ) e. CC )
70	67, 68, 69	3syl 20	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
71	62, 70	eqeltrd 2529	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
72	57	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
73		fveq2 5852	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
74	73	fveq1d 5854	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
75	74	adantl 466	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
76		fveq2 5852	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
77	76	fveq1d 5854	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
78	77	adantl 466	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
79	9, 13, 14, 15, 49, 71, 72, 75, 78	sumpair 31357	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
80	29, 62	oveq12d 6295	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
81	79, 80	eqtrd 2482	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
82	1, 81	mpteq2da 4518	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
83		prfi 7793	. . . 4 |- { 1 , 2 } e. Fin
84	83	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
85		eqid 2441	. . . . . . . . . 10 |- X = X
86	85	ax-gen 1603	. . . . . . . . 9 |- A. x X = X
87		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
88		nfcv 2603	. . . . . . . . . . . 12 |- F/_ x k
89	87, 88	nffv 5859	. . . . . . . . . . 11 |- F/_ x ( A ` k )
90		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
91	89, 90	nfeq 2614	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
92		fveq1 5851	. . . . . . . . . . 11 |- ( ( A ` k ) = F -> ( ( A ` k ) ` x ) = ( F ` x ) )
93	92	a1d 25	. . . . . . . . . 10 |- ( ( A ` k ) = F -> ( x e. X -> ( ( A ` k ) ` x ) = ( F ` x ) ) )
94	91, 93	ralrimi 2841	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
95		mpteq12f 4509	. . . . . . . . 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 ) ) )
96	86, 94, 95	sylancr 663	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
97	96	adantl 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
98		retopon 21136	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
99	38, 98	eqeltri 2525	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
100	99	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
101		cnf2 19616	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
102	34, 100, 18, 101	syl3anc 1227	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
103		ffn 5717	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
104	102, 103	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
105	90	dffn5f 5909	. . . . . . . . 9 |- ( F Fn X <-> F = ( x e. X |-> ( F ` x ) ) )
106	104, 105	sylib 196	. . . . . . . 8 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
107	106	adantr 465	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
108	97, 107	eqtr4d 2485	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
109	18	adantr 465	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
110	108, 109	eqeltrd 2529	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
111	110	adantlr 714	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
112		refsum2cnlem1.3	. . . . . . . . . . 11 |- F/_ x G
113	89, 112	nfeq 2614	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
114		fveq1 5851	. . . . . . . . . . 11 |- ( ( A ` k ) = G -> ( ( A ` k ) ` x ) = ( G ` x ) )
115	114	a1d 25	. . . . . . . . . 10 |- ( ( A ` k ) = G -> ( x e. X -> ( ( A ` k ) ` x ) = ( G ` x ) ) )
116	113, 115	ralrimi 2841	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
117		mpteq12f 4509	. . . . . . . . 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 ) ) )
118	86, 116, 117	sylancr 663	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
119	118	adantl 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
120		cnf2 19616	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
121	34, 100, 19, 120	syl3anc 1227	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
122		ffn 5717	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
123	121, 122	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
124	112	dffn5f 5909	. . . . . . . . 9 |- ( G Fn X <-> G = ( x e. X |-> ( G ` x ) ) )
125	123, 124	sylib 196	. . . . . . . 8 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
126	125	adantr 465	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
127	119, 126	eqtr4d 2485	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
128	19	adantr 465	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
129	127, 128	eqeltrd 2529	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
130	129	adantlr 714	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
131		simpr 461	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
132	18, 19	ifcld 3965	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
133	132	adantr 465	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
134	2	fvmpt2 5944	. . . . . . . 8 |- ( ( k e. { 1 , 2 } /\ if ( k = 1 , F , G ) e. ( J Cn K ) ) -> ( A ` k ) = if ( k = 1 , F , G ) )
135	131, 133, 134	syl2anc 661	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
136		iftrue 3928	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
137	135, 136	sylan9eq 2502	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( A ` k ) = F )
138	137	orcd 392	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
139	135	adantr 465	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
140		neeq2 2724	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
141	57, 140	mpbiri 233	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
142	141	necomd 2712	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
143	142	neneqd 2643	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
144	143	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
145	144	iffalsed 3933	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
146	139, 145	eqtrd 2482	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = G )
147	146	olcd 393	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
148		elpri 4030	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
149	148	adantl 466	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
150	138, 147, 149	mpjaodan 784	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
151	111, 130, 150	mpjaodan 784	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
152	1, 38, 34, 84, 151	refsumcn 31352	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
153	82, 152	eqeltrrd 2530	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 1379   = wceq 1381   F/wnf 1601   e. wcel 1802   F/_wnfc 2589   =/= wne 2636   A.wral 2791   ifcif 3922   {cpr 4012   U.cuni 4230   |-> cmpt 4491   ran crn 4986   Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Fincfn 7514   CCcc 9488   RRcr 9489   1c1 9491   + caddc 9493   2c2 10586   (,)cioo 11533   sum_csu 13482   topGenctg 14707  TopOnctopon 19262   Cn ccn 19591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cn 19594  df-cnp 19595  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691

This theorem is referenced by:  refsum2cn  31360