Theorem refsum2cnlem1 37218

Description: This is the core Lemma for refsum2cn 37219: 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 4510	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2582	. . . . . . . 8 |- F/_ k A
5		nfcv 2584	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5884	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2584	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5884	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2584	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5884	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5884	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		1cnd 9659	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
15		2cnd 10682	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
16		1ex 9638	. . . . . . . . . . 11 |- 1 e. _V
17	16	prid1 4105	. . . . . . . . . 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 3952	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
21		eqeq1 2426	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
22	21	ifbid 3931	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
23	22, 2	fvmptg 5958	. . . . . . . . . 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 667	. . . . . . . . 9 |- ( ph -> ( A ` 1 ) = if ( 1 = 1 , F , G ) )
25		eqid 2422	. . . . . . . . . 10 |- 1 = 1
26	25	iftruei 3916	. . . . . . . . 9 |- if ( 1 = 1 , F , G ) = F
27	24, 26	syl6eq 2479	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 466	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5879	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2422	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2422	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 20248	. . . . . . . . . 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 19928	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 17	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2430	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4225	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 21769	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2454	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5737	. . . . . . . . 9 |- ( ph -> ( F : U. J --> U. K <-> F : X --> RR ) )
44	33, 43	mpbid 213	. . . . . . . 8 |- ( ph -> F : X --> RR )
45	44	anim1i 570	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F : X --> RR /\ x e. X ) )
46		ffvelrn 6031	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 9629	. . . . . . 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 2510	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10681	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4106	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52	18, 19	ifcld 3952	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
53		eqeq1 2426	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
54	53	ifbid 3931	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
55	54, 2	fvmptg 5958	. . . . . . . . . 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 667	. . . . . . . . 9 |- ( ph -> ( A ` 2 ) = if ( 2 = 1 , F , G ) )
57		1ne2 10822	. . . . . . . . . . 11 |- 1 =/= 2
58	57	nesymi 2697	. . . . . . . . . 10 |- -. 2 = 1
59	58	iffalsei 3919	. . . . . . . . 9 |- if ( 2 = 1 , F , G ) = G
60	56, 59	syl6eq 2479	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
61	60	adantr 466	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
62	61	fveq1d 5879	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
63	30, 31	cnf 20248	. . . . . . . . . 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 5737	. . . . . . . . 9 |- ( ph -> ( G : U. J --> U. K <-> G : X --> RR ) )
66	64, 65	mpbid 213	. . . . . . . 8 |- ( ph -> G : X --> RR )
67	66	anim1i 570	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( G : X --> RR /\ x e. X ) )
68		ffvelrn 6031	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
69		recn 9629	. . . . . . 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 2510	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
72	57	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
73		fveq2 5877	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
74	73	fveq1d 5879	. . . . . 6 |- ( k = 1 -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
75	74	adantl 467	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 1 ) -> ( ( A ` k ) ` x ) = ( ( A ` 1 ) ` x ) )
76		fveq2 5877	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
77	76	fveq1d 5879	. . . . . 6 |- ( k = 2 -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
78	77	adantl 467	. . . . 5 |- ( ( ( ph /\ x e. X ) /\ k = 2 ) -> ( ( A ` k ) ` x ) = ( ( A ` 2 ) ` x ) )
79	9, 13, 14, 15, 49, 71, 72, 75, 78	sumpair 37216	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
80	29, 62	oveq12d 6319	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
81	79, 80	eqtrd 2463	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
82	1, 81	mpteq2da 4506	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
83		prfi 7848	. . . 4 |- { 1 , 2 } e. Fin
84	83	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
85		eqid 2422	. . . . . . . . . 10 |- X = X
86	85	ax-gen 1665	. . . . . . . . 9 |- A. x X = X
87		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
88		nfcv 2584	. . . . . . . . . . . 12 |- F/_ x k
89	87, 88	nffv 5884	. . . . . . . . . . 11 |- F/_ x ( A ` k )
90		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
91	89, 90	nfeq 2595	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
92		fveq1 5876	. . . . . . . . . . 11 |- ( ( A ` k ) = F -> ( ( A ` k ) ` x ) = ( F ` x ) )
93	92	a1d 26	. . . . . . . . . 10 |- ( ( A ` k ) = F -> ( x e. X -> ( ( A ` k ) ` x ) = ( F ` x ) ) )
94	91, 93	ralrimi 2825	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
95		mpteq12f 4497	. . . . . . . . 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 667	. . . . . . . 8 |- ( ( A ` k ) = F -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
97	96	adantl 467	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( F ` x ) ) )
98		retopon 21770	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
99	38, 98	eqeltri 2506	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
100	99	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
101		cnf2 20251	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
102	34, 100, 18, 101	syl3anc 1264	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
103		ffn 5742	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
104	102, 103	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
105	90	dffn5f 5932	. . . . . . . . 9 |- ( F Fn X <-> F = ( x e. X |-> ( F ` x ) ) )
106	104, 105	sylib 199	. . . . . . . 8 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
107	106	adantr 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = F ) -> F = ( x e. X |-> ( F ` x ) ) )
108	97, 107	eqtr4d 2466	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = F )
109	18	adantr 466	. . . . . 6 |- ( ( ph /\ ( A ` k ) = F ) -> F e. ( J Cn K ) )
110	108, 109	eqeltrd 2510	. . . . 5 |- ( ( ph /\ ( A ` k ) = F ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
111	110	adantlr 719	. . . 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 2595	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
114		fveq1 5876	. . . . . . . . . . 11 |- ( ( A ` k ) = G -> ( ( A ` k ) ` x ) = ( G ` x ) )
115	114	a1d 26	. . . . . . . . . 10 |- ( ( A ` k ) = G -> ( x e. X -> ( ( A ` k ) ` x ) = ( G ` x ) ) )
116	113, 115	ralrimi 2825	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
117		mpteq12f 4497	. . . . . . . . 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 667	. . . . . . . 8 |- ( ( A ` k ) = G -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
119	118	adantl 467	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = ( x e. X |-> ( G ` x ) ) )
120		cnf2 20251	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
121	34, 100, 19, 120	syl3anc 1264	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
122		ffn 5742	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
123	121, 122	syl 17	. . . . . . . . 9 |- ( ph -> G Fn X )
124	112	dffn5f 5932	. . . . . . . . 9 |- ( G Fn X <-> G = ( x e. X |-> ( G ` x ) ) )
125	123, 124	sylib 199	. . . . . . . 8 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
126	125	adantr 466	. . . . . . 7 |- ( ( ph /\ ( A ` k ) = G ) -> G = ( x e. X |-> ( G ` x ) ) )
127	119, 126	eqtr4d 2466	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) = G )
128	19	adantr 466	. . . . . 6 |- ( ( ph /\ ( A ` k ) = G ) -> G e. ( J Cn K ) )
129	127, 128	eqeltrd 2510	. . . . 5 |- ( ( ph /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
130	129	adantlr 719	. . . 4 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ ( A ` k ) = G ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
131		simpr 462	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> k e. { 1 , 2 } )
132	18, 19	ifcld 3952	. . . . . . . . 9 |- ( ph -> if ( k = 1 , F , G ) e. ( J Cn K ) )
133	132	adantr 466	. . . . . . . 8 |- ( ( ph /\ k e. { 1 , 2 } ) -> if ( k = 1 , F , G ) e. ( J Cn K ) )
134	2	fvmpt2 5969	. . . . . . . 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 665	. . . . . . 7 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( A ` k ) = if ( k = 1 , F , G ) )
136		iftrue 3915	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
137	135, 136	sylan9eq 2483	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( A ` k ) = F )
138	137	orcd 393	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 1 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
139	135	adantr 466	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = if ( k = 1 , F , G ) )
140		neeq2 2707	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
141	57, 140	mpbiri 236	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
142	141	necomd 2695	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
143	142	neneqd 2625	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
144	143	adantl 467	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
145	144	iffalsed 3920	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
146	139, 145	eqtrd 2463	. . . . . 6 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( A ` k ) = G )
147	146	olcd 394	. . . . 5 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
148		elpri 4016	. . . . . 6 |- ( k e. { 1 , 2 } -> ( k = 1 \/ k = 2 ) )
149	148	adantl 467	. . . . 5 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( k = 1 \/ k = 2 ) )
150	138, 147, 149	mpjaodan 793	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
151	111, 130, 150	mpjaodan 793	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
152	1, 38, 34, 84, 151	refsumcn 37211	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
153	82, 152	eqeltrrd 2511	1 |- ( ph -> ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) e. ( J Cn K ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 369   /\ wa 370   A.wal 1435   = wceq 1437   F/wnf 1663   e. wcel 1868   F/_wnfc 2570   =/= wne 2618   A.wral 2775   ifcif 3909   {cpr 3998   U.cuni 4216   |-> cmpt 4479   ran crn 4850   Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301   Fincfn 7573   CCcc 9537   RRcr 9538   1c1 9540   + caddc 9542   2c2 10659   (,)cioo 11635   sum_csu 13739   topGenctg 15323  TopOnctopon 19904   Cn ccn 20226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539  df-sum 13740  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cn 20229  df-cnp 20230  df-tx 20563  df-hmeo 20756  df-xms 21321  df-ms 21322  df-tms 21323

This theorem is referenced by:  refsum2cn  37219