Theorem infleinf 39588

Description: If any element of B can be approximated from above by members of A, then the infimum of A is smaller or equal to the infimum of B. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
infleinf.a	|- ( ph -> A C_ RR* )
infleinf.b	|- ( ph -> B C_ RR* )
infleinf.c	|- ( ( ph /\ x e. B /\ y e. RR+ ) -> E. z e. A z <_ ( x +e y ) )

Assertion

Ref	Expression
infleinf	|- ( ph -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   ph, x, y, z

Proof of Theorem infleinf

Dummy variables r w b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infleinf.a	. . . . . 6 |- ( ph -> A C_ RR* )
2		infxrcl 12163	. . . . . 6 |- ( A C_ RR* -> inf ( A , RR* , < ) e. RR* )
3	1, 2	syl 17	. . . . 5 |- ( ph -> inf ( A , RR* , < ) e. RR* )
4		pnfge 11964	. . . . 5 |- (inf ( A , RR* , < ) e. RR* -> inf ( A , RR* , < ) <_ +oo )
5	3, 4	syl 17	. . . 4 |- ( ph -> inf ( A , RR* , < ) <_ +oo )
6	5	adantr 481	. . 3 |- ( ( ph /\ B = (/) ) -> inf ( A , RR* , < ) <_ +oo )
7		infeq1 8382	. . . . . 6 |- ( B = (/) -> inf ( B , RR* , < ) = inf ( (/) , RR* , < ) )
8		xrinf0 12168	. . . . . . 7 |- inf ( (/) , RR* , < ) = +oo
9	8	a1i 11	. . . . . 6 |- ( B = (/) -> inf ( (/) , RR* , < ) = +oo )
10	7, 9	eqtrd 2656	. . . . 5 |- ( B = (/) -> inf ( B , RR* , < ) = +oo )
11	10	eqcomd 2628	. . . 4 |- ( B = (/) -> +oo = inf ( B , RR* , < ) )
12	11	adantl 482	. . 3 |- ( ( ph /\ B = (/) ) -> +oo = inf ( B , RR* , < ) )
13	6, 12	breqtrd 4679	. 2 |- ( ( ph /\ B = (/) ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
14		neqne 2802	. . . 4 |- ( -. B = (/) -> B =/= (/) )
15	14	adantl 482	. . 3 |- ( ( ph /\ -. B = (/) ) -> B =/= (/) )
16	3	adantr 481	. . . . . 6 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) e. RR* )
17		id 22	. . . . . . . . . . . . 13 |- ( r e. RR -> r e. RR )
18		2re 11090	. . . . . . . . . . . . . 14 |- 2 e. RR
19	18	a1i 11	. . . . . . . . . . . . 13 |- ( r e. RR -> 2 e. RR )
20	17, 19	resubcld 10458	. . . . . . . . . . . 12 |- ( r e. RR -> ( r - 2 ) e. RR )
21	20	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> ( r - 2 ) e. RR )
22		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> inf ( B , RR* , < ) = -oo )
23		infleinf.b	. . . . . . . . . . . . . . 15 |- ( ph -> B C_ RR* )
24		infxrunb2 39584	. . . . . . . . . . . . . . 15 |- ( B C_ RR* -> ( A. y e. RR E. x e. B x < y <-> inf ( B , RR* , < ) = -oo ) )
25	23, 24	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( A. y e. RR E. x e. B x < y <-> inf ( B , RR* , < ) = -oo ) )
26	25	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> ( A. y e. RR E. x e. B x < y <-> inf ( B , RR* , < ) = -oo ) )
27	22, 26	mpbird 247	. . . . . . . . . . . 12 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> A. y e. RR E. x e. B x < y )
28	27	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> A. y e. RR E. x e. B x < y )
29		breq2 4657	. . . . . . . . . . . . 13 |- ( y = ( r - 2 ) -> ( x < y <-> x < ( r - 2 ) ) )
30	29	rexbidv 3052	. . . . . . . . . . . 12 |- ( y = ( r - 2 ) -> ( E. x e. B x < y <-> E. x e. B x < ( r - 2 ) ) )
31	30	rspcva 3307	. . . . . . . . . . 11 |- ( ( ( r - 2 ) e. RR /\ A. y e. RR E. x e. B x < y ) -> E. x e. B x < ( r - 2 ) )
32	21, 28, 31	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> E. x e. B x < ( r - 2 ) )
33		simpl 473	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. B ) -> ph )
34		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. B ) -> x e. B )
35		1rp 11836	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR+
36	35	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. B ) -> 1 e. RR+ )
37		1ex 10035	. . . . . . . . . . . . . . . . . 18 |- 1 e. _V
38		eleq1 2689	. . . . . . . . . . . . . . . . . . . 20 |- ( y = 1 -> ( y e. RR+ <-> 1 e. RR+ ) )
39	38	3anbi3d 1405	. . . . . . . . . . . . . . . . . . 19 |- ( y = 1 -> ( ( ph /\ x e. B /\ y e. RR+ ) <-> ( ph /\ x e. B /\ 1 e. RR+ ) ) )
40		oveq2 6658	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = 1 -> ( x +e y ) = ( x +e 1 ) )
41	40	breq2d 4665	. . . . . . . . . . . . . . . . . . . 20 |- ( y = 1 -> ( z <_ ( x +e y ) <-> z <_ ( x +e 1 ) ) )
42	41	rexbidv 3052	. . . . . . . . . . . . . . . . . . 19 |- ( y = 1 -> ( E. z e. A z <_ ( x +e y ) <-> E. z e. A z <_ ( x +e 1 ) ) )
43	39, 42	imbi12d 334	. . . . . . . . . . . . . . . . . 18 |- ( y = 1 -> ( ( ( ph /\ x e. B /\ y e. RR+ ) -> E. z e. A z <_ ( x +e y ) ) <-> ( ( ph /\ x e. B /\ 1 e. RR+ ) -> E. z e. A z <_ ( x +e 1 ) ) ) )
44		infleinf.c	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. B /\ y e. RR+ ) -> E. z e. A z <_ ( x +e y ) )
45	37, 43, 44	vtocl 3259	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. B /\ 1 e. RR+ ) -> E. z e. A z <_ ( x +e 1 ) )
46	33, 34, 36, 45	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. B ) -> E. z e. A z <_ ( x +e 1 ) )
47	46	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ r e. RR ) /\ x e. B ) -> E. z e. A z <_ ( x +e 1 ) )
48	47	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> E. z e. A z <_ ( x +e 1 ) )
49		simp1l 1085	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> ph )
50	49	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> ph )
51	50, 1	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> A C_ RR* )
52	50, 23	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> B C_ RR* )
53		simp1r 1086	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> r e. RR )
54	53	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> r e. RR )
55		simp2 1062	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> x e. B )
56	55	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> x e. B )
57		simpll3 1102	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> x < ( r - 2 ) )
58		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> z e. A )
59		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> z <_ ( x +e 1 ) )
60	51, 52, 54, 56, 57, 58, 59	infleinflem2 39587	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) /\ z <_ ( x +e 1 ) ) -> z < r )
61	60	ex 450	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) /\ z e. A ) -> ( z <_ ( x +e 1 ) -> z < r ) )
62	61	reximdva 3017	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> ( E. z e. A z <_ ( x +e 1 ) -> E. z e. A z < r ) )
63	48, 62	mpd 15	. . . . . . . . . . . . 13 |- ( ( ( ph /\ r e. RR ) /\ x e. B /\ x < ( r - 2 ) ) -> E. z e. A z < r )
64	63	3exp 1264	. . . . . . . . . . . 12 |- ( ( ph /\ r e. RR ) -> ( x e. B -> ( x < ( r - 2 ) -> E. z e. A z < r ) ) )
65	64	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> ( x e. B -> ( x < ( r - 2 ) -> E. z e. A z < r ) ) )
66	65	rexlimdv 3030	. . . . . . . . . 10 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> ( E. x e. B x < ( r - 2 ) -> E. z e. A z < r ) )
67	32, 66	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ inf ( B , RR* , < ) = -oo ) /\ r e. RR ) -> E. z e. A z < r )
68	67	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> A. r e. RR E. z e. A z < r )
69		infxrunb2 39584	. . . . . . . . . 10 |- ( A C_ RR* -> ( A. r e. RR E. z e. A z < r <-> inf ( A , RR* , < ) = -oo ) )
70	1, 69	syl 17	. . . . . . . . 9 |- ( ph -> ( A. r e. RR E. z e. A z < r <-> inf ( A , RR* , < ) = -oo ) )
71	70	adantr 481	. . . . . . . 8 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> ( A. r e. RR E. z e. A z < r <-> inf ( A , RR* , < ) = -oo ) )
72	68, 71	mpbid 222	. . . . . . 7 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) = -oo )
73	72, 22	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) = inf ( B , RR* , < ) )
74	16, 73	xreqled 39546	. . . . 5 |- ( ( ph /\ inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
75	74	adantlr 751	. . . 4 |- ( ( ( ph /\ B =/= (/) ) /\ inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
76		mnfxr 10096	. . . . . . . 8 |- -oo e. RR*
77	76	a1i 11	. . . . . . 7 |- ( ph -> -oo e. RR* )
78	77	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> -oo e. RR* )
79		infxrcl 12163	. . . . . . . 8 |- ( B C_ RR* -> inf ( B , RR* , < ) e. RR* )
80	23, 79	syl 17	. . . . . . 7 |- ( ph -> inf ( B , RR* , < ) e. RR* )
81	80	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> inf ( B , RR* , < ) e. RR* )
82		mnfle 11969	. . . . . . 7 |- (inf ( B , RR* , < ) e. RR* -> -oo <_ inf ( B , RR* , < ) )
83	81, 82	syl 17	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> -oo <_ inf ( B , RR* , < ) )
84		neqne 2802	. . . . . . . 8 |- ( -. inf ( B , RR* , < ) = -oo -> inf ( B , RR* , < ) =/= -oo )
85	84	necomd 2849	. . . . . . 7 |- ( -. inf ( B , RR* , < ) = -oo -> -oo =/= inf ( B , RR* , < ) )
86	85	adantl 482	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> -oo =/= inf ( B , RR* , < ) )
87	78, 81, 83, 86	xrleneltd 39539	. . . . 5 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> -oo < inf ( B , RR* , < ) )
88	3	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) -> inf ( A , RR* , < ) e. RR* )
89	80	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) -> inf ( B , RR* , < ) e. RR* )
90		nfv 1843	. . . . . . . 8 |- F/ b ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ )
91	23	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> B C_ RR* )
92		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> B =/= (/) )
93		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ -oo < inf ( B , RR* , < ) ) -> -oo < inf ( B , RR* , < ) )
94		infxrbnd2 39585	. . . . . . . . . . . 12 |- ( B C_ RR* -> ( E. b e. RR A. x e. B b <_ x <-> -oo < inf ( B , RR* , < ) ) )
95	23, 94	syl 17	. . . . . . . . . . 11 |- ( ph -> ( E. b e. RR A. x e. B b <_ x <-> -oo < inf ( B , RR* , < ) ) )
96	95	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -oo < inf ( B , RR* , < ) ) -> ( E. b e. RR A. x e. B b <_ x <-> -oo < inf ( B , RR* , < ) ) )
97	93, 96	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ -oo < inf ( B , RR* , < ) ) -> E. b e. RR A. x e. B b <_ x )
98	97	ad4ant13 1292	. . . . . . . 8 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> E. b e. RR A. x e. B b <_ x )
99		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> w e. RR+ )
100	99	rphalfcld 11884	. . . . . . . 8 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> ( w / 2 ) e. RR+ )
101	90, 91, 92, 98, 100	infrpge 39567	. . . . . . 7 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> E. x e. B x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) )
102		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B ) -> ph )
103		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B ) -> x e. B )
104		rphalfcl 11858	. . . . . . . . . . . . . 14 |- ( w e. RR+ -> ( w / 2 ) e. RR+ )
105	104	ad2antlr 763	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B ) -> ( w / 2 ) e. RR+ )
106		ovex 6678	. . . . . . . . . . . . . 14 |- ( w / 2 ) e. _V
107		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( y = ( w / 2 ) -> ( y e. RR+ <-> ( w / 2 ) e. RR+ ) )
108	107	3anbi3d 1405	. . . . . . . . . . . . . . 15 |- ( y = ( w / 2 ) -> ( ( ph /\ x e. B /\ y e. RR+ ) <-> ( ph /\ x e. B /\ ( w / 2 ) e. RR+ ) ) )
109		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( y = ( w / 2 ) -> ( x +e y ) = ( x +e ( w / 2 ) ) )
110	109	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( y = ( w / 2 ) -> ( z <_ ( x +e y ) <-> z <_ ( x +e ( w / 2 ) ) ) )
111	110	rexbidv 3052	. . . . . . . . . . . . . . 15 |- ( y = ( w / 2 ) -> ( E. z e. A z <_ ( x +e y ) <-> E. z e. A z <_ ( x +e ( w / 2 ) ) ) )
112	108, 111	imbi12d 334	. . . . . . . . . . . . . 14 |- ( y = ( w / 2 ) -> ( ( ( ph /\ x e. B /\ y e. RR+ ) -> E. z e. A z <_ ( x +e y ) ) <-> ( ( ph /\ x e. B /\ ( w / 2 ) e. RR+ ) -> E. z e. A z <_ ( x +e ( w / 2 ) ) ) ) )
113	106, 112, 44	vtocl 3259	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. B /\ ( w / 2 ) e. RR+ ) -> E. z e. A z <_ ( x +e ( w / 2 ) ) )
114	102, 103, 105, 113	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B ) -> E. z e. A z <_ ( x +e ( w / 2 ) ) )
115	114	3adant3 1081	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) -> E. z e. A z <_ ( x +e ( w / 2 ) ) )
116		simp11l 1172	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> ph )
117	116, 1	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> A C_ RR* )
118	116, 23	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> B C_ RR* )
119		simp11 1091	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> ( ph /\ w e. RR+ ) )
120	119	simprd 479	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> w e. RR+ )
121		simp12 1092	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> x e. B )
122		simp3 1063	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) -> x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) )
123	122	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) )
124		simp2 1062	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> z e. A )
125		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> z <_ ( x +e ( w / 2 ) ) )
126	117, 118, 120, 121, 123, 124, 125	infleinflem1 39586	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) /\ z e. A /\ z <_ ( x +e ( w / 2 ) ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) )
127	126	3exp 1264	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) -> ( z e. A -> ( z <_ ( x +e ( w / 2 ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) ) ) )
128	127	rexlimdv 3030	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) -> ( E. z e. A z <_ ( x +e ( w / 2 ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) ) )
129	115, 128	mpd 15	. . . . . . . . . 10 |- ( ( ( ph /\ w e. RR+ ) /\ x e. B /\ x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) )
130	129	3exp 1264	. . . . . . . . 9 |- ( ( ph /\ w e. RR+ ) -> ( x e. B -> ( x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) ) ) )
131	130	rexlimdv 3030	. . . . . . . 8 |- ( ( ph /\ w e. RR+ ) -> ( E. x e. B x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) ) )
132	131	ad4ant14 1293	. . . . . . 7 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> ( E. x e. B x <_ (inf ( B , RR* , < ) +e ( w / 2 ) ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) ) )
133	101, 132	mpd 15	. . . . . 6 |- ( ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) /\ w e. RR+ ) -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e w ) )
134	88, 89, 133	xrlexaddrp 39568	. . . . 5 |- ( ( ( ph /\ B =/= (/) ) /\ -oo < inf ( B , RR* , < ) ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
135	87, 134	syldan 487	. . . 4 |- ( ( ( ph /\ B =/= (/) ) /\ -. inf ( B , RR* , < ) = -oo ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
136	75, 135	pm2.61dan 832	. . 3 |- ( ( ph /\ B =/= (/) ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
137	15, 136	syldan 487	. 2 |- ( ( ph /\ -. B = (/) ) -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )
138	13, 137	pm2.61dan 832	1 |- ( ph -> inf ( A , RR* , < ) <_ inf ( B , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650  infcinf 8347   RRcr 9935   1c1 9937   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   +ecxad 11944

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947

This theorem is referenced by:  ovolval5lem3  40868