Theorem refsum2cnlem1 31615

Description: This is the core Lemma for refsum2cn 31616: 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 4546	. . . . . . . . 9 |- F/_ k ( k e. { 1 , 2 } |-> if ( k = 1 , F , G ) )
4	2, 3	nfcxfr 2617	. . . . . . . 8 |- F/_ k A
5		nfcv 2619	. . . . . . . 8 |- F/_ k 1
6	4, 5	nffv 5879	. . . . . . 7 |- F/_ k ( A ` 1 )
7		nfcv 2619	. . . . . . 7 |- F/_ k x
8	6, 7	nffv 5879	. . . . . 6 |- F/_ k ( ( A ` 1 ) ` x )
9	8	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 1 ) ` x ) )
10		nfcv 2619	. . . . . . . 8 |- F/_ k 2
11	4, 10	nffv 5879	. . . . . . 7 |- F/_ k ( A ` 2 )
12	11, 7	nffv 5879	. . . . . 6 |- F/_ k ( ( A ` 2 ) ` x )
13	12	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> F/_ k ( ( A ` 2 ) ` x ) )
14		1cnd 9629	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 e. CC )
15		2cnd 10629	. . . . 5 |- ( ( ph /\ x e. X ) -> 2 e. CC )
16		1ex 9608	. . . . . . . . . . 11 |- 1 e. _V
17	16	prid1 4140	. . . . . . . . . 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 3987	. . . . . . . . . 10 |- ( ph -> if ( 1 = 1 , F , G ) e. ( J Cn K ) )
21		eqeq1 2461	. . . . . . . . . . . 12 |- ( k = 1 -> ( k = 1 <-> 1 = 1 ) )
22	21	ifbid 3966	. . . . . . . . . . 11 |- ( k = 1 -> if ( k = 1 , F , G ) = if ( 1 = 1 , F , G ) )
23	22, 2	fvmptg 5954	. . . . . . . . . 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 2457	. . . . . . . . . 10 |- 1 = 1
26	25	iftruei 3951	. . . . . . . . 9 |- if ( 1 = 1 , F , G ) = F
27	24, 26	syl6eq 2514	. . . . . . . 8 |- ( ph -> ( A ` 1 ) = F )
28	27	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 1 ) = F )
29	28	fveq1d 5874	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) = ( F ` x ) )
30		eqid 2457	. . . . . . . . . . 11 |- U. J = U. J
31		eqid 2457	. . . . . . . . . . 11 |- U. K = U. K
32	30, 31	cnf 19874	. . . . . . . . . 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 19555	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X = U. J )
36	34, 35	syl 16	. . . . . . . . . . 11 |- ( ph -> X = U. J )
37	36	eqcomd 2465	. . . . . . . . . 10 |- ( ph -> U. J = X )
38		refsum2cnlem1.6	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
39	38	unieqi 4260	. . . . . . . . . . . 12 |- U. K = U. ( topGen ` ran (,) )
40		uniretop 21395	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
41	39, 40	eqtr4i 2489	. . . . . . . . . . 11 |- U. K = RR
42	41	a1i 11	. . . . . . . . . 10 |- ( ph -> U. K = RR )
43	37, 42	feq23d 5732	. . . . . . . . 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 6030	. . . . . . 7 |- ( ( F : X --> RR /\ x e. X ) -> ( F ` x ) e. RR )
47		recn 9599	. . . . . . 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 2545	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 1 ) ` x ) e. CC )
50		2ex 10628	. . . . . . . . . . 11 |- 2 e. _V
51	50	prid2 4141	. . . . . . . . . 10 |- 2 e. { 1 , 2 }
52	18, 19	ifcld 3987	. . . . . . . . . 10 |- ( ph -> if ( 2 = 1 , F , G ) e. ( J Cn K ) )
53		eqeq1 2461	. . . . . . . . . . . 12 |- ( k = 2 -> ( k = 1 <-> 2 = 1 ) )
54	53	ifbid 3966	. . . . . . . . . . 11 |- ( k = 2 -> if ( k = 1 , F , G ) = if ( 2 = 1 , F , G ) )
55	54, 2	fvmptg 5954	. . . . . . . . . 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 10769	. . . . . . . . . . 11 |- 1 =/= 2
58	57	nesymi 2730	. . . . . . . . . 10 |- -. 2 = 1
59	58	iffalsei 3954	. . . . . . . . 9 |- if ( 2 = 1 , F , G ) = G
60	56, 59	syl6eq 2514	. . . . . . . 8 |- ( ph -> ( A ` 2 ) = G )
61	60	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A ` 2 ) = G )
62	61	fveq1d 5874	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) = ( G ` x ) )
63	30, 31	cnf 19874	. . . . . . . . . 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 5732	. . . . . . . . 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 6030	. . . . . . 7 |- ( ( G : X --> RR /\ x e. X ) -> ( G ` x ) e. RR )
69		recn 9599	. . . . . . 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 2545	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A ` 2 ) ` x ) e. CC )
72	57	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> 1 =/= 2 )
73		fveq2 5872	. . . . . . 7 |- ( k = 1 -> ( A ` k ) = ( A ` 1 ) )
74	73	fveq1d 5874	. . . . . 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 5872	. . . . . . 7 |- ( k = 2 -> ( A ` k ) = ( A ` 2 ) )
77	76	fveq1d 5874	. . . . . 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 31613	. . . 4 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) )
80	29, 62	oveq12d 6314	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A ` 1 ) ` x ) + ( ( A ` 2 ) ` x ) ) = ( ( F ` x ) + ( G ` x ) ) )
81	79, 80	eqtrd 2498	. . 3 |- ( ( ph /\ x e. X ) -> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
82	1, 81	mpteq2da 4542	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) = ( x e. X |-> ( ( F ` x ) + ( G ` x ) ) ) )
83		prfi 7813	. . . 4 |- { 1 , 2 } e. Fin
84	83	a1i 11	. . 3 |- ( ph -> { 1 , 2 } e. Fin )
85		eqid 2457	. . . . . . . . . 10 |- X = X
86	85	ax-gen 1619	. . . . . . . . 9 |- A. x X = X
87		refsum2cnlem1.1	. . . . . . . . . . . 12 |- F/_ x A
88		nfcv 2619	. . . . . . . . . . . 12 |- F/_ x k
89	87, 88	nffv 5879	. . . . . . . . . . 11 |- F/_ x ( A ` k )
90		refsum2cnlem1.2	. . . . . . . . . . 11 |- F/_ x F
91	89, 90	nfeq 2630	. . . . . . . . . 10 |- F/ x ( A ` k ) = F
92		fveq1 5871	. . . . . . . . . . 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 2857	. . . . . . . . 9 |- ( ( A ` k ) = F -> A. x e. X ( ( A ` k ) ` x ) = ( F ` x ) )
95		mpteq12f 4533	. . . . . . . . 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 21396	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
99	38, 98	eqeltri 2541	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
100	99	a1i 11	. . . . . . . . . . 11 |- ( ph -> K e. (TopOn ` RR ) )
101		cnf2 19877	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ F e. ( J Cn K ) ) -> F : X --> RR )
102	34, 100, 18, 101	syl3anc 1228	. . . . . . . . . 10 |- ( ph -> F : X --> RR )
103		ffn 5737	. . . . . . . . . 10 |- ( F : X --> RR -> F Fn X )
104	102, 103	syl 16	. . . . . . . . 9 |- ( ph -> F Fn X )
105	90	dffn5f 5928	. . . . . . . . 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 2501	. . . . . 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 2545	. . . . 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 2630	. . . . . . . . . 10 |- F/ x ( A ` k ) = G
114		fveq1 5871	. . . . . . . . . . 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 2857	. . . . . . . . 9 |- ( ( A ` k ) = G -> A. x e. X ( ( A ` k ) ` x ) = ( G ` x ) )
117		mpteq12f 4533	. . . . . . . . 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 19877	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ G e. ( J Cn K ) ) -> G : X --> RR )
121	34, 100, 19, 120	syl3anc 1228	. . . . . . . . . 10 |- ( ph -> G : X --> RR )
122		ffn 5737	. . . . . . . . . 10 |- ( G : X --> RR -> G Fn X )
123	121, 122	syl 16	. . . . . . . . 9 |- ( ph -> G Fn X )
124	112	dffn5f 5928	. . . . . . . . 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 2501	. . . . . 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 2545	. . . . 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 3987	. . . . . . . . 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 5964	. . . . . . . 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 3950	. . . . . . 7 |- ( k = 1 -> if ( k = 1 , F , G ) = F )
137	135, 136	sylan9eq 2518	. . . . . 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 2740	. . . . . . . . . . . 12 |- ( k = 2 -> ( 1 =/= k <-> 1 =/= 2 ) )
141	57, 140	mpbiri 233	. . . . . . . . . . 11 |- ( k = 2 -> 1 =/= k )
142	141	necomd 2728	. . . . . . . . . 10 |- ( k = 2 -> k =/= 1 )
143	142	neneqd 2659	. . . . . . . . 9 |- ( k = 2 -> -. k = 1 )
144	143	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> -. k = 1 )
145	144	iffalsed 3955	. . . . . . 7 |- ( ( ( ph /\ k e. { 1 , 2 } ) /\ k = 2 ) -> if ( k = 1 , F , G ) = G )
146	139, 145	eqtrd 2498	. . . . . 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 4052	. . . . . 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 786	. . . 4 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( ( A ` k ) = F \/ ( A ` k ) = G ) )
151	111, 130, 150	mpjaodan 786	. . 3 |- ( ( ph /\ k e. { 1 , 2 } ) -> ( x e. X |-> ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
152	1, 38, 34, 84, 151	refsumcn 31608	. 2 |- ( ph -> ( x e. X |-> sum_ k e. { 1 , 2 } ( ( A ` k ) ` x ) ) e. ( J Cn K ) )
153	82, 152	eqeltrrd 2546	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 1393   = wceq 1395   F/wnf 1617   e. wcel 1819   F/_wnfc 2605   =/= wne 2652   A.wral 2807   ifcif 3944   {cpr 4034   U.cuni 4251   |-> cmpt 4515   ran crn 5009   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   RRcr 9508   1c1 9510   + caddc 9512   2c2 10606   (,)cioo 11554   sum_csu 13520   topGenctg 14855  TopOnctopon 19522   Cn ccn 19852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cn 19855  df-cnp 19856  df-tx 20189  df-hmeo 20382  df-xms 20949  df-ms 20950  df-tms 20951

This theorem is referenced by:  refsum2cn  31616