Theorem ovnsubadd2lem 40859

Description: (voln^* ` X ) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovnsubadd2lem.x	|- ( ph -> X e. Fin )
ovnsubadd2lem.a	|- ( ph -> A C_ ( RR ^m X ) )
ovnsubadd2lem.b	|- ( ph -> B C_ ( RR ^m X ) )
ovnsubadd2lem.c	|- C = ( n e. NN |-> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) )

Assertion

Ref	Expression
ovnsubadd2lem	|- ( ph -> ( (voln* ` X ) ` ( A u. B ) ) <_ ( ( (voln* ` X ) ` A ) +e ( (voln* ` X ) ` B ) ) )

Distinct variable groups:   A, n   B, n   C, n   n, X   ph, n

Proof of Theorem ovnsubadd2lem

Step	Hyp	Ref	Expression
1		ovnsubadd2lem.x	. . 3 |- ( ph -> X e. Fin )
2		iftrue 4092	. . . . . . . 8 |- ( n = 1 -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = A )
3	2	adantl 482	. . . . . . 7 |- ( ( ph /\ n = 1 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = A )
4		ovexd 6680	. . . . . . . . . 10 |- ( ph -> ( RR ^m X ) e. _V )
5		ovnsubadd2lem.a	. . . . . . . . . 10 |- ( ph -> A C_ ( RR ^m X ) )
6	4, 5	ssexd 4805	. . . . . . . . 9 |- ( ph -> A e. _V )
7	6, 5	elpwd 4167	. . . . . . . 8 |- ( ph -> A e. ~P ( RR ^m X ) )
8	7	adantr 481	. . . . . . 7 |- ( ( ph /\ n = 1 ) -> A e. ~P ( RR ^m X ) )
9	3, 8	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ n = 1 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
10	9	adantlr 751	. . . . 5 |- ( ( ( ph /\ n e. NN ) /\ n = 1 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
11		simpl 473	. . . . . . . . . . 11 |- ( ( -. n = 1 /\ n = 2 ) -> -. n = 1 )
12	11	iffalsed 4097	. . . . . . . . . 10 |- ( ( -. n = 1 /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = if ( n = 2 , B , (/) ) )
13		simpr 477	. . . . . . . . . . 11 |- ( ( -. n = 1 /\ n = 2 ) -> n = 2 )
14	13	iftrued 4094	. . . . . . . . . 10 |- ( ( -. n = 1 /\ n = 2 ) -> if ( n = 2 , B , (/) ) = B )
15	12, 14	eqtrd 2656	. . . . . . . . 9 |- ( ( -. n = 1 /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = B )
16	15	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ -. n = 1 ) /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = B )
17		ovnsubadd2lem.b	. . . . . . . . . . 11 |- ( ph -> B C_ ( RR ^m X ) )
18	4, 17	ssexd 4805	. . . . . . . . . 10 |- ( ph -> B e. _V )
19	18, 17	elpwd 4167	. . . . . . . . 9 |- ( ph -> B e. ~P ( RR ^m X ) )
20	19	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ -. n = 1 ) /\ n = 2 ) -> B e. ~P ( RR ^m X ) )
21	16, 20	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ -. n = 1 ) /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
22	21	adantllr 755	. . . . . 6 |- ( ( ( ( ph /\ n e. NN ) /\ -. n = 1 ) /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
23		simpl 473	. . . . . . . . . 10 |- ( ( -. n = 1 /\ -. n = 2 ) -> -. n = 1 )
24	23	iffalsed 4097	. . . . . . . . 9 |- ( ( -. n = 1 /\ -. n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = if ( n = 2 , B , (/) ) )
25		simpr 477	. . . . . . . . . 10 |- ( ( -. n = 1 /\ -. n = 2 ) -> -. n = 2 )
26	25	iffalsed 4097	. . . . . . . . 9 |- ( ( -. n = 1 /\ -. n = 2 ) -> if ( n = 2 , B , (/) ) = (/) )
27	24, 26	eqtrd 2656	. . . . . . . 8 |- ( ( -. n = 1 /\ -. n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = (/) )
28		0elpw 4834	. . . . . . . . 9 |- (/) e. ~P ( RR ^m X )
29	28	a1i 11	. . . . . . . 8 |- ( ( -. n = 1 /\ -. n = 2 ) -> (/) e. ~P ( RR ^m X ) )
30	27, 29	eqeltrd 2701	. . . . . . 7 |- ( ( -. n = 1 /\ -. n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
31	30	adantll 750	. . . . . 6 |- ( ( ( ( ph /\ n e. NN ) /\ -. n = 1 ) /\ -. n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
32	22, 31	pm2.61dan 832	. . . . 5 |- ( ( ( ph /\ n e. NN ) /\ -. n = 1 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
33	10, 32	pm2.61dan 832	. . . 4 |- ( ( ph /\ n e. NN ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. ~P ( RR ^m X ) )
34		ovnsubadd2lem.c	. . . 4 |- C = ( n e. NN |-> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) )
35	33, 34	fmptd 6385	. . 3 |- ( ph -> C : NN --> ~P ( RR ^m X ) )
36	1, 35	ovnsubadd 40786	. 2 |- ( ph -> ( (voln* ` X ) ` U_ n e. NN ( C ` n ) ) <_ (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) )
37		eldifi 3732	. . . . . . . . . . 11 |- ( n e. ( NN \ { 1 , 2 } ) -> n e. NN )
38	37	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> n e. NN )
39		eldifn 3733	. . . . . . . . . . . . . 14 |- ( n e. ( NN \ { 1 , 2 } ) -> -. n e. { 1 , 2 } )
40		vex 3203	. . . . . . . . . . . . . . . . 17 |- n e. _V
41	40	a1i 11	. . . . . . . . . . . . . . . 16 |- ( -. n e. { 1 , 2 } -> n e. _V )
42		id 22	. . . . . . . . . . . . . . . 16 |- ( -. n e. { 1 , 2 } -> -. n e. { 1 , 2 } )
43	41, 42	nelpr1 39262	. . . . . . . . . . . . . . 15 |- ( -. n e. { 1 , 2 } -> n =/= 1 )
44	43	neneqd 2799	. . . . . . . . . . . . . 14 |- ( -. n e. { 1 , 2 } -> -. n = 1 )
45	39, 44	syl 17	. . . . . . . . . . . . 13 |- ( n e. ( NN \ { 1 , 2 } ) -> -. n = 1 )
46	41, 42	nelpr2 39261	. . . . . . . . . . . . . . 15 |- ( -. n e. { 1 , 2 } -> n =/= 2 )
47	46	neneqd 2799	. . . . . . . . . . . . . 14 |- ( -. n e. { 1 , 2 } -> -. n = 2 )
48	39, 47	syl 17	. . . . . . . . . . . . 13 |- ( n e. ( NN \ { 1 , 2 } ) -> -. n = 2 )
49	45, 48, 27	syl2anc 693	. . . . . . . . . . . 12 |- ( n e. ( NN \ { 1 , 2 } ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = (/) )
50		0ex 4790	. . . . . . . . . . . . 13 |- (/) e. _V
51	50	a1i 11	. . . . . . . . . . . 12 |- ( n e. ( NN \ { 1 , 2 } ) -> (/) e. _V )
52	49, 51	eqeltrd 2701	. . . . . . . . . . 11 |- ( n e. ( NN \ { 1 , 2 } ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. _V )
53	52	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. _V )
54	34	fvmpt2 6291	. . . . . . . . . 10 |- ( ( n e. NN /\ if ( n = 1 , A , if ( n = 2 , B , (/) ) ) e. _V ) -> ( C ` n ) = if ( n = 1 , A , if ( n = 2 , B , (/) ) ) )
55	38, 53, 54	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> ( C ` n ) = if ( n = 1 , A , if ( n = 2 , B , (/) ) ) )
56	49	adantl 482	. . . . . . . . 9 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = (/) )
57	55, 56	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> ( C ` n ) = (/) )
58	57	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. n e. ( NN \ { 1 , 2 } ) ( C ` n ) = (/) )
59		nfcv 2764	. . . . . . . 8 |- F/_ n ( NN \ { 1 , 2 } )
60	59	iunxdif3 4606	. . . . . . 7 |- ( A. n e. ( NN \ { 1 , 2 } ) ( C ` n ) = (/) -> U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) = U_ n e. NN ( C ` n ) )
61	58, 60	syl 17	. . . . . 6 |- ( ph -> U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) = U_ n e. NN ( C ` n ) )
62	61	eqcomd 2628	. . . . 5 |- ( ph -> U_ n e. NN ( C ` n ) = U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) )
63		1nn 11031	. . . . . . . . . 10 |- 1 e. NN
64		2nn 11185	. . . . . . . . . 10 |- 2 e. NN
65	63, 64	pm3.2i 471	. . . . . . . . 9 |- ( 1 e. NN /\ 2 e. NN )
66		prssi 4353	. . . . . . . . 9 |- ( ( 1 e. NN /\ 2 e. NN ) -> { 1 , 2 } C_ NN )
67	65, 66	ax-mp 5	. . . . . . . 8 |- { 1 , 2 } C_ NN
68		dfss4 3858	. . . . . . . 8 |- ( { 1 , 2 } C_ NN <-> ( NN \ ( NN \ { 1 , 2 } ) ) = { 1 , 2 } )
69	67, 68	mpbi 220	. . . . . . 7 |- ( NN \ ( NN \ { 1 , 2 } ) ) = { 1 , 2 }
70		iuneq1 4534	. . . . . . 7 |- ( ( NN \ ( NN \ { 1 , 2 } ) ) = { 1 , 2 } -> U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) = U_ n e. { 1 , 2 } ( C ` n ) )
71	69, 70	ax-mp 5	. . . . . 6 |- U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) = U_ n e. { 1 , 2 } ( C ` n )
72	71	a1i 11	. . . . 5 |- ( ph -> U_ n e. ( NN \ ( NN \ { 1 , 2 } ) ) ( C ` n ) = U_ n e. { 1 , 2 } ( C ` n ) )
73		fveq2 6191	. . . . . . . . 9 |- ( n = 1 -> ( C ` n ) = ( C ` 1 ) )
74		fveq2 6191	. . . . . . . . 9 |- ( n = 2 -> ( C ` n ) = ( C ` 2 ) )
75	73, 74	iunxprg 4607	. . . . . . . 8 |- ( ( 1 e. NN /\ 2 e. NN ) -> U_ n e. { 1 , 2 } ( C ` n ) = ( ( C ` 1 ) u. ( C ` 2 ) ) )
76	63, 64, 75	mp2an 708	. . . . . . 7 |- U_ n e. { 1 , 2 } ( C ` n ) = ( ( C ` 1 ) u. ( C ` 2 ) )
77	76	a1i 11	. . . . . 6 |- ( ph -> U_ n e. { 1 , 2 } ( C ` n ) = ( ( C ` 1 ) u. ( C ` 2 ) ) )
78	34	a1i 11	. . . . . . . 8 |- ( ph -> C = ( n e. NN |-> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) ) )
79	63	a1i 11	. . . . . . . 8 |- ( ph -> 1 e. NN )
80	78, 3, 79, 6	fvmptd 6288	. . . . . . 7 |- ( ph -> ( C ` 1 ) = A )
81		id 22	. . . . . . . . . . . . 13 |- ( n = 2 -> n = 2 )
82		1ne2 11240	. . . . . . . . . . . . . . 15 |- 1 =/= 2
83	82	necomi 2848	. . . . . . . . . . . . . 14 |- 2 =/= 1
84	83	a1i 11	. . . . . . . . . . . . 13 |- ( n = 2 -> 2 =/= 1 )
85	81, 84	eqnetrd 2861	. . . . . . . . . . . 12 |- ( n = 2 -> n =/= 1 )
86	85	neneqd 2799	. . . . . . . . . . 11 |- ( n = 2 -> -. n = 1 )
87	86	iffalsed 4097	. . . . . . . . . 10 |- ( n = 2 -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = if ( n = 2 , B , (/) ) )
88		iftrue 4092	. . . . . . . . . 10 |- ( n = 2 -> if ( n = 2 , B , (/) ) = B )
89	87, 88	eqtrd 2656	. . . . . . . . 9 |- ( n = 2 -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = B )
90	89	adantl 482	. . . . . . . 8 |- ( ( ph /\ n = 2 ) -> if ( n = 1 , A , if ( n = 2 , B , (/) ) ) = B )
91	64	a1i 11	. . . . . . . 8 |- ( ph -> 2 e. NN )
92	78, 90, 91, 18	fvmptd 6288	. . . . . . 7 |- ( ph -> ( C ` 2 ) = B )
93	80, 92	uneq12d 3768	. . . . . 6 |- ( ph -> ( ( C ` 1 ) u. ( C ` 2 ) ) = ( A u. B ) )
94		eqidd 2623	. . . . . 6 |- ( ph -> ( A u. B ) = ( A u. B ) )
95	77, 93, 94	3eqtrd 2660	. . . . 5 |- ( ph -> U_ n e. { 1 , 2 } ( C ` n ) = ( A u. B ) )
96	62, 72, 95	3eqtrd 2660	. . . 4 |- ( ph -> U_ n e. NN ( C ` n ) = ( A u. B ) )
97	96	fveq2d 6195	. . 3 |- ( ph -> ( (voln* ` X ) ` U_ n e. NN ( C ` n ) ) = ( (voln* ` X ) ` ( A u. B ) ) )
98		nfv 1843	. . . . . 6 |- F/ n ph
99		nnex 11026	. . . . . . 7 |- NN e. _V
100	99	a1i 11	. . . . . 6 |- ( ph -> NN e. _V )
101	67	a1i 11	. . . . . 6 |- ( ph -> { 1 , 2 } C_ NN )
102	1	adantr 481	. . . . . . 7 |- ( ( ph /\ n e. { 1 , 2 } ) -> X e. Fin )
103		simpl 473	. . . . . . . 8 |- ( ( ph /\ n e. { 1 , 2 } ) -> ph )
104	101	sselda 3603	. . . . . . . 8 |- ( ( ph /\ n e. { 1 , 2 } ) -> n e. NN )
105	35	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( C ` n ) e. ~P ( RR ^m X ) )
106		elpwi 4168	. . . . . . . . 9 |- ( ( C ` n ) e. ~P ( RR ^m X ) -> ( C ` n ) C_ ( RR ^m X ) )
107	105, 106	syl 17	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( C ` n ) C_ ( RR ^m X ) )
108	103, 104, 107	syl2anc 693	. . . . . . 7 |- ( ( ph /\ n e. { 1 , 2 } ) -> ( C ` n ) C_ ( RR ^m X ) )
109	102, 108	ovncl 40781	. . . . . 6 |- ( ( ph /\ n e. { 1 , 2 } ) -> ( (voln* ` X ) ` ( C ` n ) ) e. ( 0 [,] +oo ) )
110	57	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> ( (voln* ` X ) ` ( C ` n ) ) = ( (voln* ` X ) ` (/) ) )
111	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> X e. Fin )
112	111	ovn0 40780	. . . . . . 7 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> ( (voln* ` X ) ` (/) ) = 0 )
113	110, 112	eqtrd 2656	. . . . . 6 |- ( ( ph /\ n e. ( NN \ { 1 , 2 } ) ) -> ( (voln* ` X ) ` ( C ` n ) ) = 0 )
114	98, 100, 101, 109, 113	sge0ss 40629	. . . . 5 |- ( ph -> (Σ^ ` ( n e. { 1 , 2 } |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) = (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) )
115	114	eqcomd 2628	. . . 4 |- ( ph -> (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) = (Σ^ ` ( n e. { 1 , 2 } |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) )
116	80, 5	eqsstrd 3639	. . . . . 6 |- ( ph -> ( C ` 1 ) C_ ( RR ^m X ) )
117	1, 116	ovncl 40781	. . . . 5 |- ( ph -> ( (voln* ` X ) ` ( C ` 1 ) ) e. ( 0 [,] +oo ) )
118	92, 17	eqsstrd 3639	. . . . . 6 |- ( ph -> ( C ` 2 ) C_ ( RR ^m X ) )
119	1, 118	ovncl 40781	. . . . 5 |- ( ph -> ( (voln* ` X ) ` ( C ` 2 ) ) e. ( 0 [,] +oo ) )
120	73	fveq2d 6195	. . . . 5 |- ( n = 1 -> ( (voln* ` X ) ` ( C ` n ) ) = ( (voln* ` X ) ` ( C ` 1 ) ) )
121	74	fveq2d 6195	. . . . 5 |- ( n = 2 -> ( (voln* ` X ) ` ( C ` n ) ) = ( (voln* ` X ) ` ( C ` 2 ) ) )
122	82	a1i 11	. . . . 5 |- ( ph -> 1 =/= 2 )
123	79, 91, 117, 119, 120, 121, 122	sge0pr 40611	. . . 4 |- ( ph -> (Σ^ ` ( n e. { 1 , 2 } |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) = ( ( (voln* ` X ) ` ( C ` 1 ) ) +e ( (voln* ` X ) ` ( C ` 2 ) ) ) )
124	80	fveq2d 6195	. . . . 5 |- ( ph -> ( (voln* ` X ) ` ( C ` 1 ) ) = ( (voln* ` X ) ` A ) )
125	92	fveq2d 6195	. . . . 5 |- ( ph -> ( (voln* ` X ) ` ( C ` 2 ) ) = ( (voln* ` X ) ` B ) )
126	124, 125	oveq12d 6668	. . . 4 |- ( ph -> ( ( (voln* ` X ) ` ( C ` 1 ) ) +e ( (voln* ` X ) ` ( C ` 2 ) ) ) = ( ( (voln* ` X ) ` A ) +e ( (voln* ` X ) ` B ) ) )
127	115, 123, 126	3eqtrd 2660	. . 3 |- ( ph -> (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) = ( ( (voln* ` X ) ` A ) +e ( (voln* ` X ) ` B ) ) )
128	97, 127	breq12d 4666	. 2 |- ( ph -> ( ( (voln* ` X ) ` U_ n e. NN ( C ` n ) ) <_ (Σ^ ` ( n e. NN |-> ( (voln* ` X ) ` ( C ` n ) ) ) ) <-> ( (voln* ` X ) ` ( A u. B ) ) <_ ( ( (voln* ` X ) ` A ) +e ( (voln* ` X ) ` B ) ) ) )
129	36, 128	mpbid 222	1 |- ( ph -> ( (voln* ` X ) ` ( A u. B ) ) <_ ( ( (voln* ` X ) ` A ) +e ( (voln* ` X ) ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {cpr 4179   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   <_ cle 10075   NNcn 11020   2c2 11070   +ecxad 11944  Σ^{^}csumge0 40579  voln^*covoln 40750

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-cc 9257  ax-ac2 9285  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  ax-mulf 10016

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-disj 4621  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-tpos 7352  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-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417  df-prod 14636  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-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  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-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580  df-ovoln 40751

This theorem is referenced by:  ovnsubadd2  40860