Theorem refsum2cnlem1 39196

Description: This is the core Lemma for refsum2cn 39197: 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 4747	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2762	. . . . . . . 8 |- F/_ k A
5		nfcv 2764	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 6198	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2764	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 6198	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2764	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 6198	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 6198	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		1cnd 10056	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
15		2cnd 11093	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
16		1ex 10035	. . . . . . . . . . 11 |- 1 e. _V
17	16	prid1 4297	. . . . . . . . . 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 4131	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
21		eqeq1 2626	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
22	21	ifbid 4108	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
23	22, 2	fvmptg 6280	. . . . . . . . . 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 695	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
25		eqid 2622	. . . . . . . . . 10 |- 1 = 1
26	25	iftruei 4093	. . . . . . . . 9 |- if ( 1 = 1 , F , G ) = F
27	24, 26	syl6eq 2672	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 6193	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2622	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2622	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 21050	. . . . . . . . . 10 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
33	18, 32	syl 17	. . . . . . . . 9 |- ( ph -> F : U. J --> U. K )
34		refsum2cnlem1.7	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` X ) )
35		toponuni 20719	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 17	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2628	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4445	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 22566	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2647	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 6040	. . . . . . . . 9 |- ( ph -> ( F : U. J --> U. K <-> F : X --> RR ) )
44	33, 43	mpbid 222	. . . . . . . 8 |- ( ph -> F : X --> RR )
45	44	anim1i 592	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F : X --> RR /\ x e. X ) )
46		ffvelrn 6357	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 10026	. . . . . . 7 |- ( ( F ` x ) e. RR -> ( F ` x ) e. CC )
48	45, 46, 47	3syl 18	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
49	29, 48	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 11092	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4298	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52	18, 19	ifcld 4131	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
53		eqeq1 2626	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
54	53	ifbid 4108	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
55	54, 2	fvmptg 6280	. . . . . . . . . 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 695	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
57		1ne2 11240	. . . . . . . . . . 11 |- 1 =/= 2
58	57	nesymi 2851	. . . . . . . . . 10 |- -. 2 = 1
59	58	iffalsei 4096	. . . . . . . . 9 |- if ( 2 = 1 , F , G ) = G
60	56, 59	syl6eq 2672	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
61	60	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
62	61	fveq1d 6193	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
63	30, 31	cnf 21050	. . . . . . . . . 10 |- ( G e. ( J Cn K ) -> G : U. J --> U. K )
64	19, 63	syl 17	. . . . . . . . 9 |- ( ph -> G : U. J --> U. K )
65	37, 42	feq23d 6040	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
66	64, 65	mpbid 222	. . . . . . . 8 |- ( ph -> G : X --> RR )
67	66	anim1i 592	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
68		ffvelrn 6357	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
69		recn 10026	. . . . . . 7 |- ( ( G ` x ) e. RR -> ( G ` x ) e. CC )
70	67, 68, 69	3syl 18	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
71	62, 70	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
72	57	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
73		fveq2 6191	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
74	73	fveq1d 6193	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
75	74	adantl 482	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
76		fveq2 6191	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
77	76	fveq1d 6193	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
78	77	adantl 482	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
79	9, 13, 14, 15, 49, 71, 72, 75, 78	sumpair 39194	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
80	29, 62	oveq12d 6668	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
81	79, 80	eqtrd 2656	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
82	1, 81	mpteq2da 4743	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
83		prfi 8235	. . . 4 |- { 1 , 2 } e. Fin
84	83	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
85		eqid 2622	. . . . . . . . . 10 |- X = X
86	85	ax-gen 1722	. . . . . . . . 9 |- A. x X = X
87		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
88		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x k
89	87, 88	nffv 6198	. . . . . . . . . . 11 |- F/_ x ( A ` k )
90		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
91	89, 90	nfeq 2776	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
92		fveq1 6190	. . . . . . . . . . 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 2957	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
95		mpteq12f 4731	. . . . . . . . 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 695	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
97	96	adantl 482	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
98		retopon 22567	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
99	38, 98	eqeltri 2697	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
100	99	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
101		cnf2 21053	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
102	34, 100, 18, 101	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
103		ffn 6045	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
104	102, 103	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
105	90	dffn5f 6252	. . . . . . . . 9 |- ( F Fn X <-> F = ( x e. X |-> ( F ` x ) ) )
106	104, 105	sylib 208	. . . . . . . 8 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
107	106	adantr 481	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
108	97, 107	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
109	18	adantr 481	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
110	108, 109	eqeltrd 2701	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
111	110	adantlr 751	. . . 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 2776	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
114		fveq1 6190	. . . . . . . . . . 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 2957	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
117		mpteq12f 4731	. . . . . . . . 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 695	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
119	118	adantl 482	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
120		cnf2 21053	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
121	34, 100, 19, 120	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
122		ffn 6045	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
123	121, 122	syl 17	. . . . . . . . 9 |- ( ph -> G Fn X )
124	112	dffn5f 6252	. . . . . . . . 9 |- ( G Fn X <-> G = ( x e. X |-> ( G ` x ) ) )
125	123, 124	sylib 208	. . . . . . . 8 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
126	125	adantr 481	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
127	119, 126	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
128	19	adantr 481	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
129	127, 128	eqeltrd 2701	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
130	129	adantlr 751	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
131		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
132	18, 19	ifcld 4131	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
133	132	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
134	2	fvmpt2 6291	. . . . . . . 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 693	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
136		iftrue 4092	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
137	135, 136	sylan9eq 2676	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( A ` k ) = F )
138	137	orcd 407	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
139	135	adantr 481	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
140		neeq2 2857	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
141	57, 140	mpbiri 248	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
142	141	necomd 2849	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
143	142	neneqd 2799	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
144	143	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
145	144	iffalsed 4097	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
146	139, 145	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = G )
147	146	olcd 408	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
148		elpri 4197	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
149	148	adantl 482	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
150	138, 147, 149	mpjaodan 827	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
151	111, 130, 150	mpjaodan 827	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
152	1, 38, 34, 84, 151	refsumcn 39189	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
153	82, 152	eqeltrrd 2702	1 |- ( ph -> ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) e. ( J Cn K ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   ifcif 4086   {cpr 4179   U.cuni 4436   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   2c2 11070   (,)cioo 12175   sum_csu 14416   topGenctg 16098  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  refsum2cn  39197