Theorem infxr 39583

Description: The infimum of a set of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
infxr.x	|- F/ x ph
infxr.y	|- F/ y ph
infxr.a	|- ( ph -> A C_ RR* )
infxr.b	|- ( ph -> B e. RR* )
infxr.n	|- ( ph -> A. x e. A -. x < B )
infxr.e	|- ( ph -> A. x e. RR ( B < x -> E. y e. A y < x ) )

Assertion

Ref	Expression
infxr	|- ( ph -> inf ( A , RR* , < ) = B )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem infxr

Step	Hyp	Ref	Expression
1		infxr.b	. 2 |- ( ph -> B e. RR* )
2		infxr.n	. 2 |- ( ph -> A. x e. A -. x < B )
3		infxr.x	. . 3 |- F/ x ph
4		infxr.e	. . . . . . 7 |- ( ph -> A. x e. RR ( B < x -> E. y e. A y < x ) )
5	4	r19.21bi 2932	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( B < x -> E. y e. A y < x ) )
6	5	adantlr 751	. . . . 5 |- ( ( ( ph /\ x e. RR* ) /\ x e. RR ) -> ( B < x -> E. y e. A y < x ) )
7		simplll 798	. . . . . . 7 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> ph )
8		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x e. RR* )
9		simplr 792	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> -. x e. RR )
10		mnfxr 10096	. . . . . . . . . . 11 |- -oo e. RR*
11	10	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo e. RR* )
12		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> x e. RR* )
13	1	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> B e. RR* )
14		mnfle 11969	. . . . . . . . . . . . 13 |- ( B e. RR* -> -oo <_ B )
15	1, 14	syl 17	. . . . . . . . . . . 12 |- ( ph -> -oo <_ B )
16	15	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo <_ B )
17		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> B < x )
18	11, 13, 12, 16, 17	xrlelttrd 11991	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo < x )
19	11, 12, 18	xrgtned 39538	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> x =/= -oo )
20	19	adantlr 751	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x =/= -oo )
21	8, 9, 20	xrnmnfpnf 39256	. . . . . . 7 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x = +oo )
22		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> B < x )
23		simpl 473	. . . . . . . . . . . 12 |- ( ( ph /\ B = -oo ) -> ph )
24		id 22	. . . . . . . . . . . . . 14 |- ( B = -oo -> B = -oo )
25		1re 10039	. . . . . . . . . . . . . . 15 |- 1 e. RR
26		mnflt 11957	. . . . . . . . . . . . . . 15 |- ( 1 e. RR -> -oo < 1 )
27	25, 26	ax-mp 5	. . . . . . . . . . . . . 14 |- -oo < 1
28	24, 27	syl6eqbr 4692	. . . . . . . . . . . . 13 |- ( B = -oo -> B < 1 )
29	28	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ B = -oo ) -> B < 1 )
30		1red 10055	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
31		breq2 4657	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( B < x <-> B < 1 ) )
32		breq2 4657	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( y < x <-> y < 1 ) )
33	32	rexbidv 3052	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( E. y e. A y < x <-> E. y e. A y < 1 ) )
34	31, 33	imbi12d 334	. . . . . . . . . . . . . 14 |- ( x = 1 -> ( ( B < x -> E. y e. A y < x ) <-> ( B < 1 -> E. y e. A y < 1 ) ) )
35	34	rspcva 3307	. . . . . . . . . . . . 13 |- ( ( 1 e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) -> ( B < 1 -> E. y e. A y < 1 ) )
36	30, 4, 35	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( B < 1 -> E. y e. A y < 1 ) )
37	23, 29, 36	sylc 65	. . . . . . . . . . 11 |- ( ( ph /\ B = -oo ) -> E. y e. A y < 1 )
38	37	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> E. y e. A y < 1 )
39		infxr.y	. . . . . . . . . . . . 13 |- F/ y ph
40		nfv 1843	. . . . . . . . . . . . 13 |- F/ y x = +oo
41	39, 40	nfan 1828	. . . . . . . . . . . 12 |- F/ y ( ph /\ x = +oo )
42		infxr.a	. . . . . . . . . . . . . . . . 17 |- ( ph -> A C_ RR* )
43	42	sselda 3603	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. A ) -> y e. RR* )
44	43	ad4ant13 1292	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y e. RR* )
45	25	rexri 10097	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
46	45	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> 1 e. RR* )
47		id 22	. . . . . . . . . . . . . . . . . 18 |- ( x = +oo -> x = +oo )
48		pnfxr 10092	. . . . . . . . . . . . . . . . . 18 |- +oo e. RR*
49	47, 48	syl6eqel 2709	. . . . . . . . . . . . . . . . 17 |- ( x = +oo -> x e. RR* )
50	49	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x = +oo ) -> x e. RR* )
51	50	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> x e. RR* )
52		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y < 1 )
53		ltpnf 11954	. . . . . . . . . . . . . . . . . . . 20 |- ( 1 e. RR -> 1 < +oo )
54	25, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- 1 < +oo
55	54	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( x = +oo -> 1 < +oo )
56	47	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( x = +oo -> +oo = x )
57	55, 56	breqtrd 4679	. . . . . . . . . . . . . . . . 17 |- ( x = +oo -> 1 < x )
58	57	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x = +oo ) -> 1 < x )
59	58	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> 1 < x )
60	44, 46, 51, 52, 59	xrlttrd 11990	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y < x )
61	60	ex 450	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x = +oo ) /\ y e. A ) -> ( y < 1 -> y < x ) )
62	61	ex 450	. . . . . . . . . . . 12 |- ( ( ph /\ x = +oo ) -> ( y e. A -> ( y < 1 -> y < x ) ) )
63	41, 62	reximdai 3012	. . . . . . . . . . 11 |- ( ( ph /\ x = +oo ) -> ( E. y e. A y < 1 -> E. y e. A y < x ) )
64	63	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> ( E. y e. A y < 1 -> E. y e. A y < x ) )
65	38, 64	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> E. y e. A y < x )
66	65	3adantl3 1219	. . . . . . . 8 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ B = -oo ) -> E. y e. A y < x )
67	1	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. B = -oo ) -> B e. RR* )
68	67	3ad2antl1 1223	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B e. RR* )
69	24	necon3bi 2820	. . . . . . . . . . . . . 14 |- ( -. B = -oo -> B =/= -oo )
70	69	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B =/= -oo )
71	48	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> +oo e. RR* )
72		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( x = +oo /\ B < x ) -> B < x )
73		simpl 473	. . . . . . . . . . . . . . . . 17 |- ( ( x = +oo /\ B < x ) -> x = +oo )
74	72, 73	breqtrd 4679	. . . . . . . . . . . . . . . 16 |- ( ( x = +oo /\ B < x ) -> B < +oo )
75	74	3adant1 1079	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x = +oo /\ B < x ) -> B < +oo )
76	75	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B < +oo )
77	68, 71, 76	xrltned 39573	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B =/= +oo )
78	68, 70, 77	xrred 39581	. . . . . . . . . . . 12 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B e. RR )
79	25	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> 1 e. RR )
80	78, 79	readdcld 10069	. . . . . . . . . . 11 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) e. RR )
81	4	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ -. B = -oo ) -> A. x e. RR ( B < x -> E. y e. A y < x ) )
82	81	3ad2antl1 1223	. . . . . . . . . . 11 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> A. x e. RR ( B < x -> E. y e. A y < x ) )
83	80, 82	jca 554	. . . . . . . . . 10 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( ( B + 1 ) e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) )
84	78	ltp1d 10954	. . . . . . . . . 10 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B < ( B + 1 ) )
85		breq2 4657	. . . . . . . . . . . 12 |- ( x = ( B + 1 ) -> ( B < x <-> B < ( B + 1 ) ) )
86		breq2 4657	. . . . . . . . . . . . 13 |- ( x = ( B + 1 ) -> ( y < x <-> y < ( B + 1 ) ) )
87	86	rexbidv 3052	. . . . . . . . . . . 12 |- ( x = ( B + 1 ) -> ( E. y e. A y < x <-> E. y e. A y < ( B + 1 ) ) )
88	85, 87	imbi12d 334	. . . . . . . . . . 11 |- ( x = ( B + 1 ) -> ( ( B < x -> E. y e. A y < x ) <-> ( B < ( B + 1 ) -> E. y e. A y < ( B + 1 ) ) ) )
89	88	rspcva 3307	. . . . . . . . . 10 |- ( ( ( B + 1 ) e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) -> ( B < ( B + 1 ) -> E. y e. A y < ( B + 1 ) ) )
90	83, 84, 89	sylc 65	. . . . . . . . 9 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> E. y e. A y < ( B + 1 ) )
91		nfv 1843	. . . . . . . . . . . 12 |- F/ y B < x
92	39, 40, 91	nf3an 1831	. . . . . . . . . . 11 |- F/ y ( ph /\ x = +oo /\ B < x )
93		nfv 1843	. . . . . . . . . . 11 |- F/ y -. B = -oo
94	92, 93	nfan 1828	. . . . . . . . . 10 |- F/ y ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo )
95	43	3ad2antl1 1223	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ y e. A ) -> y e. RR* )
96	95	ad4ant13 1292	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y e. RR* )
97	80	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( B + 1 ) e. RR )
98	97	rexrd 10089	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( B + 1 ) e. RR* )
99	98	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> ( B + 1 ) e. RR* )
100	50	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( ph /\ x = +oo /\ B < x ) -> x e. RR* )
101	100	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> x e. RR* )
102		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y < ( B + 1 ) )
103	80	ltpnfd 11955	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) < +oo )
104	56	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( x = +oo /\ -. B = -oo ) -> +oo = x )
105	104	3ad2antl2 1224	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> +oo = x )
106	103, 105	breqtrd 4679	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) < x )
107	106	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> ( B + 1 ) < x )
108	96, 99, 101, 102, 107	xrlttrd 11990	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y < x )
109	108	ex 450	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( y < ( B + 1 ) -> y < x ) )
110	109	ex 450	. . . . . . . . . 10 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( y e. A -> ( y < ( B + 1 ) -> y < x ) ) )
111	94, 110	reximdai 3012	. . . . . . . . 9 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( E. y e. A y < ( B + 1 ) -> E. y e. A y < x ) )
112	90, 111	mpd 15	. . . . . . . 8 |- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> E. y e. A y < x )
113	66, 112	pm2.61dan 832	. . . . . . 7 |- ( ( ph /\ x = +oo /\ B < x ) -> E. y e. A y < x )
114	7, 21, 22, 113	syl3anc 1326	. . . . . 6 |- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> E. y e. A y < x )
115	114	ex 450	. . . . 5 |- ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) -> ( B < x -> E. y e. A y < x ) )
116	6, 115	pm2.61dan 832	. . . 4 |- ( ( ph /\ x e. RR* ) -> ( B < x -> E. y e. A y < x ) )
117	116	ex 450	. . 3 |- ( ph -> ( x e. RR* -> ( B < x -> E. y e. A y < x ) ) )
118	3, 117	ralrimi 2957	. 2 |- ( ph -> A. x e. RR* ( B < x -> E. y e. A y < x ) )
119		xrltso 11974	. . . . 5 |- < Or RR*
120	119	a1i 11	. . . 4 |- ( T. -> < Or RR* )
121	120	eqinf 8390	. . 3 |- ( T. -> ( ( B e. RR* /\ A. x e. A -. x < B /\ A. x e. RR* ( B < x -> E. y e. A y < x ) ) -> inf ( A , RR* , < ) = B ) )
122	121	trud 1493	. 2 |- ( ( B e. RR* /\ A. x e. A -. x < B /\ A. x e. RR* ( B < x -> E. y e. A y < x ) ) -> inf ( A , RR* , < ) = B )
123	1, 2, 118, 122	syl3anc 1326	1 |- ( ph -> inf ( A , RR* , < ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   Or wor 5034  (class class class)co 6650  infcinf 8347   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075

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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-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

This theorem is referenced by:  infxrunb2  39584