Theorem suplesup 37637

Description: If any element of A can be approximated from below by members of B, then the supremum of A is smaller or equal to the supremum of B. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

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

Assertion

Ref	Expression
suplesup	|- ( ph -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )

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

Allowed substitution hints:   ph( y)   A( y)

Proof of Theorem suplesup

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

Step	Hyp	Ref	Expression
1		suplesup.a	. . . . . 6 |- ( ph -> A C_ RR )
2		ressxr 9715	. . . . . 6 |- RR C_ RR*
3	1, 2	syl6ss 3456	. . . . 5 |- ( ph -> A C_ RR* )
4		supxrcl 11634	. . . . 5 |- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
5	3, 4	syl 17	. . . 4 |- ( ph -> sup ( A , RR* , < ) e. RR* )
6	5	adantr 471	. . 3 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) e. RR* )
7		eqidd 2463	. . . 4 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> +oo = +oo )
8		simpr 467	. . . 4 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) = +oo )
9		peano2re 9837	. . . . . . . . . 10 |- ( w e. RR -> ( w + 1 ) e. RR )
10	9	adantl 472	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> ( w + 1 ) e. RR )
11	3	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> A C_ RR* )
12		supxrunb2 11640	. . . . . . . . . . . 12 |- ( A C_ RR* -> ( A. r e. RR E. x e. A r < x <-> sup ( A , RR* , < ) = +oo ) )
13	11, 12	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> ( A. r e. RR E. x e. A r < x <-> sup ( A , RR* , < ) = +oo ) )
14	8, 13	mpbird 240	. . . . . . . . . 10 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> A. r e. RR E. x e. A r < x )
15	14	adantr 471	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> A. r e. RR E. x e. A r < x )
16		breq1 4421	. . . . . . . . . . 11 |- ( r = ( w + 1 ) -> ( r < x <-> ( w + 1 ) < x ) )
17	16	rexbidv 2913	. . . . . . . . . 10 |- ( r = ( w + 1 ) -> ( E. x e. A r < x <-> E. x e. A ( w + 1 ) < x ) )
18	17	rspcva 3160	. . . . . . . . 9 |- ( ( ( w + 1 ) e. RR /\ A. r e. RR E. x e. A r < x ) -> E. x e. A ( w + 1 ) < x )
19	10, 15, 18	syl2anc 671	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> E. x e. A ( w + 1 ) < x )
20		1rp 11340	. . . . . . . . . . . . . . . 16 |- 1 e. RR+
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. A ) -> 1 e. RR+ )
22		suplesup.c	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. A A. y e. RR+ E. z e. B ( x - y ) < z )
23	22	r19.21bi 2769	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. A ) -> A. y e. RR+ E. z e. B ( x - y ) < z )
24		oveq2 6328	. . . . . . . . . . . . . . . . . 18 |- ( y = 1 -> ( x - y ) = ( x - 1 ) )
25	24	breq1d 4428	. . . . . . . . . . . . . . . . 17 |- ( y = 1 -> ( ( x - y ) < z <-> ( x - 1 ) < z ) )
26	25	rexbidv 2913	. . . . . . . . . . . . . . . 16 |- ( y = 1 -> ( E. z e. B ( x - y ) < z <-> E. z e. B ( x - 1 ) < z ) )
27	26	rspcva 3160	. . . . . . . . . . . . . . 15 |- ( ( 1 e. RR+ /\ A. y e. RR+ E. z e. B ( x - y ) < z ) -> E. z e. B ( x - 1 ) < z )
28	21, 23, 27	syl2anc 671	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> E. z e. B ( x - 1 ) < z )
29	28	adantlr 726	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> E. z e. B ( x - 1 ) < z )
30	29	3adant3 1034	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> E. z e. B ( x - 1 ) < z )
31		nfv 1772	. . . . . . . . . . . . 13 |- F/ z ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x )
32		simp11r 1126	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> w e. RR )
33	2, 32	sseldi 3442	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> w e. RR* )
34	1	sselda 3444	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. A ) -> x e. RR )
35		1red 9689	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. A ) -> 1 e. RR )
36	34, 35	resubcld 10080	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. A ) -> ( x - 1 ) e. RR )
37	36	adantlr 726	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> ( x - 1 ) e. RR )
38	37	3adant3 1034	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> ( x - 1 ) e. RR )
39	38	3ad2ant1 1035	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> ( x - 1 ) e. RR )
40	2, 39	sseldi 3442	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> ( x - 1 ) e. RR* )
41		suplesup.b	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> B C_ RR* )
42	41	sselda 3444	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. B ) -> z e. RR* )
43	42	adantlr 726	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR ) /\ z e. B ) -> z e. RR* )
44	43	3ad2antl1 1176	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B ) -> z e. RR* )
45	44	3adant3 1034	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> z e. RR* )
46		simp3 1016	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> ( w + 1 ) < x )
47		simp1r 1039	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> w e. RR )
48		1red 9689	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> 1 e. RR )
49	34	adantlr 726	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ w e. RR ) /\ x e. A ) -> x e. RR )
50	49	3adant3 1034	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> x e. RR )
51	47, 48, 50	ltaddsubd 10246	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> ( ( w + 1 ) < x <-> w < ( x - 1 ) ) )
52	46, 51	mpbid 215	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> w < ( x - 1 ) )
53	52	3ad2ant1 1035	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> w < ( x - 1 ) )
54		simp3 1016	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> ( x - 1 ) < z )
55	33, 40, 45, 53, 54	xrlttrd 11490	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) /\ z e. B /\ ( x - 1 ) < z ) -> w < z )
56	55	3exp 1214	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> ( z e. B -> ( ( x - 1 ) < z -> w < z ) ) )
57	31, 56	reximdai 2868	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> ( E. z e. B ( x - 1 ) < z -> E. z e. B w < z ) )
58	30, 57	mpd 15	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. RR ) /\ x e. A /\ ( w + 1 ) < x ) -> E. z e. B w < z )
59	58	3exp 1214	. . . . . . . . . 10 |- ( ( ph /\ w e. RR ) -> ( x e. A -> ( ( w + 1 ) < x -> E. z e. B w < z ) ) )
60	59	adantlr 726	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> ( x e. A -> ( ( w + 1 ) < x -> E. z e. B w < z ) ) )
61	60	rexlimdv 2889	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> ( E. x e. A ( w + 1 ) < x -> E. z e. B w < z ) )
62	19, 61	mpd 15	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> E. z e. B w < z )
63	2	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR C_ RR* )
64	63	sselda 3444	. . . . . . . . . . . 12 |- ( ( ph /\ w e. RR ) -> w e. RR* )
65	64	ad2antrr 737	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. RR ) /\ z e. B ) /\ w < z ) -> w e. RR* )
66	43	adantr 471	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. RR ) /\ z e. B ) /\ w < z ) -> z e. RR* )
67		simpr 467	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w e. RR ) /\ z e. B ) /\ w < z ) -> w < z )
68	65, 66, 67	xrltled 37559	. . . . . . . . . 10 |- ( ( ( ( ph /\ w e. RR ) /\ z e. B ) /\ w < z ) -> w <_ z )
69	68	ex 440	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ z e. B ) -> ( w < z -> w <_ z ) )
70	69	adantllr 730	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) /\ z e. B ) -> ( w < z -> w <_ z ) )
71	70	reximdva 2874	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> ( E. z e. B w < z -> E. z e. B w <_ z ) )
72	62, 71	mpd 15	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) = +oo ) /\ w e. RR ) -> E. z e. B w <_ z )
73	72	ralrimiva 2814	. . . . 5 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> A. w e. RR E. z e. B w <_ z )
74		supxrunb1 11639	. . . . . . 7 |- ( B C_ RR* -> ( A. w e. RR E. z e. B w <_ z <-> sup ( B , RR* , < ) = +oo ) )
75	41, 74	syl 17	. . . . . 6 |- ( ph -> ( A. w e. RR E. z e. B w <_ z <-> sup ( B , RR* , < ) = +oo ) )
76	75	adantr 471	. . . . 5 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> ( A. w e. RR E. z e. B w <_ z <-> sup ( B , RR* , < ) = +oo ) )
77	73, 76	mpbid 215	. . . 4 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> sup ( B , RR* , < ) = +oo )
78	7, 8, 77	3eqtr4d 2506	. . 3 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) = sup ( B , RR* , < ) )
79	6, 78	xreqled 37628	. 2 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
80		supeq1 7990	. . . . . . 7 |- ( A = (/) -> sup ( A , RR* , < ) = sup ( (/) , RR* , < ) )
81		xrsup0 11643	. . . . . . . 8 |- sup ( (/) , RR* , < ) = -oo
82	81	a1i 11	. . . . . . 7 |- ( A = (/) -> sup ( (/) , RR* , < ) = -oo )
83	80, 82	eqtrd 2496	. . . . . 6 |- ( A = (/) -> sup ( A , RR* , < ) = -oo )
84	83	adantl 472	. . . . 5 |- ( ( ph /\ A = (/) ) -> sup ( A , RR* , < ) = -oo )
85		supxrcl 11634	. . . . . . . 8 |- ( B C_ RR* -> sup ( B , RR* , < ) e. RR* )
86	41, 85	syl 17	. . . . . . 7 |- ( ph -> sup ( B , RR* , < ) e. RR* )
87		mnfle 11469	. . . . . . 7 |- ( sup ( B , RR* , < ) e. RR* -> -oo <_ sup ( B , RR* , < ) )
88	86, 87	syl 17	. . . . . 6 |- ( ph -> -oo <_ sup ( B , RR* , < ) )
89	88	adantr 471	. . . . 5 |- ( ( ph /\ A = (/) ) -> -oo <_ sup ( B , RR* , < ) )
90	84, 89	eqbrtrd 4439	. . . 4 |- ( ( ph /\ A = (/) ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
91	90	adantlr 726	. . 3 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ A = (/) ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
92		simpll 765	. . . 4 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> ph )
93	1	adantr 471	. . . . . . . 8 |- ( ( ph /\ -. A = (/) ) -> A C_ RR )
94		neqne 2643	. . . . . . . . 9 |- ( -. A = (/) -> A =/= (/) )
95	94	adantl 472	. . . . . . . 8 |- ( ( ph /\ -. A = (/) ) -> A =/= (/) )
96		supxrgtmnf 11649	. . . . . . . 8 |- ( ( A C_ RR /\ A =/= (/) ) -> -oo < sup ( A , RR* , < ) )
97	93, 95, 96	syl2anc 671	. . . . . . 7 |- ( ( ph /\ -. A = (/) ) -> -oo < sup ( A , RR* , < ) )
98	97	adantlr 726	. . . . . 6 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> -oo < sup ( A , RR* , < ) )
99		simpr 467	. . . . . . . . 9 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. sup ( A , RR* , < ) = +oo )
100		simpl 463	. . . . . . . . . 10 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ph )
101		nltpnft 11495	. . . . . . . . . 10 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) )
102	100, 5, 101	3syl 18	. . . . . . . . 9 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) )
103	99, 102	mtbid 306	. . . . . . . 8 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. -. sup ( A , RR* , < ) < +oo )
104		notnot2 117	. . . . . . . 8 |- ( -. -. sup ( A , RR* , < ) < +oo -> sup ( A , RR* , < ) < +oo )
105	103, 104	syl 17	. . . . . . 7 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) < +oo )
106	105	adantr 471	. . . . . 6 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> sup ( A , RR* , < ) < +oo )
107	98, 106	jca 539	. . . . 5 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) )
108	92, 5	syl 17	. . . . . 6 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> sup ( A , RR* , < ) e. RR* )
109		xrrebnd 11497	. . . . . 6 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) )
110	108, 109	syl 17	. . . . 5 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) )
111	107, 110	mpbird 240	. . . 4 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> sup ( A , RR* , < ) e. RR )
112		nfv 1772	. . . . 5 |- F/ w ( ph /\ sup ( A , RR* , < ) e. RR )
113	41	adantr 471	. . . . 5 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> B C_ RR* )
114		simpr 467	. . . . 5 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) e. RR )
115	114	adantr 471	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> sup ( A , RR* , < ) e. RR )
116		simpr 467	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> w e. RR+ )
117	116	rphalfcld 11387	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( w / 2 ) e. RR+ )
118	115, 117	ltsubrpd 11404	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - ( w / 2 ) ) < sup ( A , RR* , < ) )
119	3	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> A C_ RR* )
120		rpre 11342	. . . . . . . . . . . 12 |- ( w e. RR+ -> w e. RR )
121		2re 10712	. . . . . . . . . . . . 13 |- 2 e. RR
122	121	a1i 11	. . . . . . . . . . . 12 |- ( w e. RR+ -> 2 e. RR )
123		2ne0 10735	. . . . . . . . . . . . 13 |- 2 =/= 0
124	123	a1i 11	. . . . . . . . . . . 12 |- ( w e. RR+ -> 2 =/= 0 )
125	120, 122, 124	redivcld 10468	. . . . . . . . . . 11 |- ( w e. RR+ -> ( w / 2 ) e. RR )
126	125	adantl 472	. . . . . . . . . 10 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( w / 2 ) e. RR )
127	115, 126	resubcld 10080	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - ( w / 2 ) ) e. RR )
128	2, 127	sseldi 3442	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - ( w / 2 ) ) e. RR* )
129		supxrlub 11645	. . . . . . . 8 |- ( ( A C_ RR* /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) e. RR* ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) < sup ( A , RR* , < ) <-> E. x e. A ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) )
130	119, 128, 129	syl2anc 671	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) < sup ( A , RR* , < ) <-> E. x e. A ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) )
131	118, 130	mpbid 215	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> E. x e. A ( sup ( A , RR* , < ) - ( w / 2 ) ) < x )
132		rphalfcl 11361	. . . . . . . . . . . 12 |- ( w e. RR+ -> ( w / 2 ) e. RR+ )
133	132	3ad2ant2 1036	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR+ /\ x e. A ) -> ( w / 2 ) e. RR+ )
134	23	3adant2 1033	. . . . . . . . . . 11 |- ( ( ph /\ w e. RR+ /\ x e. A ) -> A. y e. RR+ E. z e. B ( x - y ) < z )
135		oveq2 6328	. . . . . . . . . . . . . 14 |- ( y = ( w / 2 ) -> ( x - y ) = ( x - ( w / 2 ) ) )
136	135	breq1d 4428	. . . . . . . . . . . . 13 |- ( y = ( w / 2 ) -> ( ( x - y ) < z <-> ( x - ( w / 2 ) ) < z ) )
137	136	rexbidv 2913	. . . . . . . . . . . 12 |- ( y = ( w / 2 ) -> ( E. z e. B ( x - y ) < z <-> E. z e. B ( x - ( w / 2 ) ) < z ) )
138	137	rspcva 3160	. . . . . . . . . . 11 |- ( ( ( w / 2 ) e. RR+ /\ A. y e. RR+ E. z e. B ( x - y ) < z ) -> E. z e. B ( x - ( w / 2 ) ) < z )
139	133, 134, 138	syl2anc 671	. . . . . . . . . 10 |- ( ( ph /\ w e. RR+ /\ x e. A ) -> E. z e. B ( x - ( w / 2 ) ) < z )
140	139	ad5ant134 1263	. . . . . . . . 9 |- ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) -> E. z e. B ( x - ( w / 2 ) ) < z )
141		recn 9660	. . . . . . . . . . . . . . . . . 18 |- ( sup ( A , RR* , < ) e. RR -> sup ( A , RR* , < ) e. CC )
142	141	adantr 471	. . . . . . . . . . . . . . . . 17 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> sup ( A , RR* , < ) e. CC )
143	120	recnd 9700	. . . . . . . . . . . . . . . . . . 19 |- ( w e. RR+ -> w e. CC )
144	143	adantl 472	. . . . . . . . . . . . . . . . . 18 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> w e. CC )
145	144	halfcld 10891	. . . . . . . . . . . . . . . . 17 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> ( w / 2 ) e. CC )
146	142, 145, 145	subsub4d 10048	. . . . . . . . . . . . . . . 16 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) = ( sup ( A , RR* , < ) - ( ( w / 2 ) + ( w / 2 ) ) ) )
147	143	2halvesd 10892	. . . . . . . . . . . . . . . . . 18 |- ( w e. RR+ -> ( ( w / 2 ) + ( w / 2 ) ) = w )
148	147	oveq2d 6336	. . . . . . . . . . . . . . . . 17 |- ( w e. RR+ -> ( sup ( A , RR* , < ) - ( ( w / 2 ) + ( w / 2 ) ) ) = ( sup ( A , RR* , < ) - w ) )
149	148	adantl 472	. . . . . . . . . . . . . . . 16 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - ( ( w / 2 ) + ( w / 2 ) ) ) = ( sup ( A , RR* , < ) - w ) )
150	146, 149	eqtr2d 2497	. . . . . . . . . . . . . . 15 |- ( ( sup ( A , RR* , < ) e. RR /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - w ) = ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) )
151	150	adantll 725	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( sup ( A , RR* , < ) - w ) = ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) )
152	151	adantr 471	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) -> ( sup ( A , RR* , < ) - w ) = ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) )
153	152	ad3antrrr 741	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( sup ( A , RR* , < ) - w ) = ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) )
154	127, 126	resubcld 10080	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) e. RR )
155	154	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) e. RR )
156	155	ad3antrrr 741	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) e. RR )
157	2, 156	sseldi 3442	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) e. RR* )
158	120, 49	sylanl2 661	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR+ ) /\ x e. A ) -> x e. RR )
159	125	ad2antlr 738	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. RR+ ) /\ x e. A ) -> ( w / 2 ) e. RR )
160	158, 159	resubcld 10080	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w e. RR+ ) /\ x e. A ) -> ( x - ( w / 2 ) ) e. RR )
161	160	adantllr 730	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) -> ( x - ( w / 2 ) ) e. RR )
162	161	ad3antrrr 741	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( x - ( w / 2 ) ) e. RR )
163	2, 162	sseldi 3442	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( x - ( w / 2 ) ) e. RR* )
164		simp-6l 785	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ph )
165		simplr 767	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> z e. B )
166	164, 165, 42	syl2anc 671	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> z e. RR* )
167		simp-6r 786	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> sup ( A , RR* , < ) e. RR )
168	120	ad5antlr 746	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> w e. RR )
169	168	rehalfcld 10893	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( w / 2 ) e. RR )
170	167, 169	resubcld 10080	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( sup ( A , RR* , < ) - ( w / 2 ) ) e. RR )
171		simp-4r 782	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> x e. A )
172	164, 171, 34	syl2anc 671	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> x e. RR )
173		simpllr 774	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( sup ( A , RR* , < ) - ( w / 2 ) ) < x )
174	170, 172, 169, 173	ltsub1dd 10258	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) < ( x - ( w / 2 ) ) )
175		simpr 467	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( x - ( w / 2 ) ) < z )
176	157, 163, 166, 174, 175	xrlttrd 11490	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) - ( w / 2 ) ) < z )
177	153, 176	eqbrtrd 4439	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) /\ ( x - ( w / 2 ) ) < z ) -> ( sup ( A , RR* , < ) - w ) < z )
178	177	ex 440	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) /\ z e. B ) -> ( ( x - ( w / 2 ) ) < z -> ( sup ( A , RR* , < ) - w ) < z ) )
179	178	reximdva 2874	. . . . . . . . 9 |- ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) -> ( E. z e. B ( x - ( w / 2 ) ) < z -> E. z e. B ( sup ( A , RR* , < ) - w ) < z ) )
180	140, 179	mpd 15	. . . . . . . 8 |- ( ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) /\ ( sup ( A , RR* , < ) - ( w / 2 ) ) < x ) -> E. z e. B ( sup ( A , RR* , < ) - w ) < z )
181	180	ex 440	. . . . . . 7 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) /\ x e. A ) -> ( ( sup ( A , RR* , < ) - ( w / 2 ) ) < x -> E. z e. B ( sup ( A , RR* , < ) - w ) < z ) )
182	181	rexlimdva 2891	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> ( E. x e. A ( sup ( A , RR* , < ) - ( w / 2 ) ) < x -> E. z e. B ( sup ( A , RR* , < ) - w ) < z ) )
183	131, 182	mpd 15	. . . . 5 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ w e. RR+ ) -> E. z e. B ( sup ( A , RR* , < ) - w ) < z )
184	112, 113, 114, 183	supxrgere 37631	. . . 4 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
185	92, 111, 184	syl2anc 671	. . 3 |- ( ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) /\ -. A = (/) ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
186	91, 185	pm2.61dan 805	. 2 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )
187	79, 186	pm2.61dan 805	1 |- ( ph -> sup ( A , RR* , < ) <_ sup ( B , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   E.wrex 2750   C_ wss 3416   (/)c0 3743   class class class wbr 4418  (class class class)co 6320   supcsup 7985   CCcc 9568   RRcr 9569   0cc0 9570   1c1 9571   + caddc 9573   +oocpnf 9703   -oocmnf 9704   RR*cxr 9705   < clt 9706   <_ cle 9707   - cmin 9891   / cdiv 10302   2c2 10692   RR+crp 11336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-po 4777  df-so 4778  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-sup 7987  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-2 10701  df-rp 11337

This theorem is referenced by: (None)