Theorem supmul1 10598

Description: The supremum function distributes over multiplication, in the sense that A x. ( sup B ) = sup ( A x. B ), where A x. B is shorthand for { A x. b | b e. B } and is defined as C below. This is the simple version, with only one set argument; see supmul 10601 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)

Hypotheses

Ref	Expression
supmul1.1	|- C = { z | E. v e. B z = ( A x. v ) }
supmul1.2	|- ( ph <-> ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) )

Assertion

Ref	Expression
supmul1	|- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Distinct variable groups:   v, A, x, z   v, B, x, y, z   x, C

Allowed substitution hints:   ph( x, y, z, v)   A( y)   C( y, z, v)

Proof of Theorem supmul1

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

Step	Hyp	Ref	Expression
1		vex 3034	. . . . . . . 8 |- w e. _V
2		oveq2 6316	. . . . . . . . . . 11 |- ( v = b -> ( A x. v ) = ( A x. b ) )
3	2	eqeq2d 2481	. . . . . . . . . 10 |- ( v = b -> ( z = ( A x. v ) <-> z = ( A x. b ) ) )
4	3	cbvrexv 3006	. . . . . . . . 9 |- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) )
5		eqeq1 2475	. . . . . . . . . 10 |- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) )
6	5	rexbidv 2892	. . . . . . . . 9 |- ( z = w -> ( E. b e. B z = ( A x. b ) <-> E. b e. B w = ( A x. b ) ) )
7	4, 6	syl5bb 265	. . . . . . . 8 |- ( z = w -> ( E. v e. B z = ( A x. v ) <-> E. b e. B w = ( A x. b ) ) )
8		supmul1.1	. . . . . . . 8 |- C = { z | E. v e. B z = ( A x. v ) }
9	1, 7, 8	elab2 3176	. . . . . . 7 |- ( w e. C <-> E. b e. B w = ( A x. b ) )
10		supmul1.2	. . . . . . . . . . . . 13 |- ( ph <-> ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) )
11		simpr 468	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
12	10, 11	sylbi 200	. . . . . . . . . . . 12 |- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
13	12	simp1d 1042	. . . . . . . . . . 11 |- ( ph -> B C_ RR )
14	13	sselda 3418	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b e. RR )
15		suprcl 10591	. . . . . . . . . . . 12 |- ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) -> sup ( B , RR , < ) e. RR )
16	12, 15	syl 17	. . . . . . . . . . 11 |- ( ph -> sup ( B , RR , < ) e. RR )
17	16	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR )
18		simpl1 1033	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> A e. RR )
19	10, 18	sylbi 200	. . . . . . . . . . . 12 |- ( ph -> A e. RR )
20		simpl2 1034	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> 0 <_ A )
21	10, 20	sylbi 200	. . . . . . . . . . . 12 |- ( ph -> 0 <_ A )
22	19, 21	jca 541	. . . . . . . . . . 11 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
23	22	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) )
24		suprub 10592	. . . . . . . . . . 11 |- ( ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) /\ b e. B ) -> b <_ sup ( B , RR , < ) )
25	12, 24	sylan 479	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) )
26		lemul2a 10482	. . . . . . . . . 10 |- ( ( ( b e. RR /\ sup ( B , RR , < ) e. RR /\ ( A e. RR /\ 0 <_ A ) ) /\ b <_ sup ( B , RR , < ) ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
27	14, 17, 23, 25, 26	syl31anc 1295	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
28		breq1 4398	. . . . . . . . 9 |- ( w = ( A x. b ) -> ( w <_ ( A x. sup ( B , RR , < ) ) <-> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) )
29	27, 28	syl5ibrcom 230	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
30	29	rexlimdva 2871	. . . . . . 7 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
31	9, 30	syl5bi 225	. . . . . 6 |- ( ph -> ( w e. C -> w <_ ( A x. sup ( B , RR , < ) ) ) )
32	31	ralrimiv 2808	. . . . 5 |- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) )
33	19	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B ) -> A e. RR )
34	33, 14	remulcld 9689	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR )
35		eleq1a 2544	. . . . . . . . . . 11 |- ( ( A x. b ) e. RR -> ( w = ( A x. b ) -> w e. RR ) )
36	34, 35	syl 17	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w e. RR ) )
37	36	rexlimdva 2871	. . . . . . . . 9 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w e. RR ) )
38	9, 37	syl5bi 225	. . . . . . . 8 |- ( ph -> ( w e. C -> w e. RR ) )
39	38	ssrdv 3424	. . . . . . 7 |- ( ph -> C C_ RR )
40		simpr2 1037	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> B =/= (/) )
41	10, 40	sylbi 200	. . . . . . . . 9 |- ( ph -> B =/= (/) )
42		ovex 6336	. . . . . . . . . . 11 |- ( A x. b ) e. _V
43	42	isseti 3037	. . . . . . . . . 10 |- E. w w = ( A x. b )
44	43	rgenw 2768	. . . . . . . . 9 |- A. b e. B E. w w = ( A x. b )
45		r19.2z 3849	. . . . . . . . 9 |- ( ( B =/= (/) /\ A. b e. B E. w w = ( A x. b ) ) -> E. b e. B E. w w = ( A x. b ) )
46	41, 44, 45	sylancl 675	. . . . . . . 8 |- ( ph -> E. b e. B E. w w = ( A x. b ) )
47	9	exbii 1726	. . . . . . . . 9 |- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) )
48		n0 3732	. . . . . . . . 9 |- ( C =/= (/) <-> E. w w e. C )
49		rexcom4 3053	. . . . . . . . 9 |- ( E. b e. B E. w w = ( A x. b ) <-> E. w E. b e. B w = ( A x. b ) )
50	47, 48, 49	3bitr4i 285	. . . . . . . 8 |- ( C =/= (/) <-> E. b e. B E. w w = ( A x. b ) )
51	46, 50	sylibr 217	. . . . . . 7 |- ( ph -> C =/= (/) )
52	19, 16	remulcld 9689	. . . . . . . 8 |- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR )
53		breq2 4399	. . . . . . . . . 10 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( A x. sup ( B , RR , < ) ) ) )
54	53	ralbidv 2829	. . . . . . . . 9 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( A. w e. C w <_ x <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
55	54	rspcev 3136	. . . . . . . 8 |- ( ( ( A x. sup ( B , RR , < ) ) e. RR /\ A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) -> E. x e. RR A. w e. C w <_ x )
56	52, 32, 55	syl2anc 673	. . . . . . 7 |- ( ph -> E. x e. RR A. w e. C w <_ x )
57	39, 51, 56	3jca 1210	. . . . . 6 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
58		suprleub 10595	. . . . . 6 |- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ ( A x. sup ( B , RR , < ) ) e. RR ) -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
59	57, 52, 58	syl2anc 673	. . . . 5 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
60	32, 59	mpbird 240	. . . 4 |- ( ph -> sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) )
61		simpr 468	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
62		suprcl 10591	. . . . . . . . . 10 |- ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) -> sup ( C , RR , < ) e. RR )
63	57, 62	syl 17	. . . . . . . . 9 |- ( ph -> sup ( C , RR , < ) e. RR )
64	63	adantr 472	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR )
65	16	adantr 472	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR )
66	19	adantr 472	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR )
67		n0 3732	. . . . . . . . . . . 12 |- ( B =/= (/) <-> E. b b e. B )
68		0red 9662	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 e. RR )
69		simpl3 1035	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> A. x e. B 0 <_ x )
70	10, 69	sylbi 200	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. B 0 <_ x )
71		breq2 4399	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( 0 <_ x <-> 0 <_ b ) )
72	71	rspccva 3135	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b )
73	70, 72	sylan 479	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 <_ b )
74	68, 14, 17, 73, 25	letrd 9809	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> 0 <_ sup ( B , RR , < ) )
75	74	ex 441	. . . . . . . . . . . . 13 |- ( ph -> ( b e. B -> 0 <_ sup ( B , RR , < ) ) )
76	75	exlimdv 1787	. . . . . . . . . . . 12 |- ( ph -> ( E. b b e. B -> 0 <_ sup ( B , RR , < ) ) )
77	67, 76	syl5bi 225	. . . . . . . . . . 11 |- ( ph -> ( B =/= (/) -> 0 <_ sup ( B , RR , < ) ) )
78	41, 77	mpd 15	. . . . . . . . . 10 |- ( ph -> 0 <_ sup ( B , RR , < ) )
79	78	adantr 472	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) )
80		0red 9662	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 e. RR )
81	38	imp 436	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w e. RR )
82	63	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR )
83	21	adantr 472	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ b e. B ) -> 0 <_ A )
84	33, 14, 83, 73	mulge0d 10211	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) )
85		breq2 4399	. . . . . . . . . . . . . . . . . . . 20 |- ( w = ( A x. b ) -> ( 0 <_ w <-> 0 <_ ( A x. b ) ) )
86	84, 85	syl5ibrcom 230	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> 0 <_ w ) )
87	86	rexlimdva 2871	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( E. b e. B w = ( A x. b ) -> 0 <_ w ) )
88	9, 87	syl5bi 225	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( w e. C -> 0 <_ w ) )
89	88	imp 436	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 <_ w )
90		suprub 10592	. . . . . . . . . . . . . . . . 17 |- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
91	57, 90	sylan 479	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
92	80, 81, 82, 89, 91	letrd 9809	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. C ) -> 0 <_ sup ( C , RR , < ) )
93	92	ex 441	. . . . . . . . . . . . . 14 |- ( ph -> ( w e. C -> 0 <_ sup ( C , RR , < ) ) )
94	93	exlimdv 1787	. . . . . . . . . . . . 13 |- ( ph -> ( E. w w e. C -> 0 <_ sup ( C , RR , < ) ) )
95	48, 94	syl5bi 225	. . . . . . . . . . . 12 |- ( ph -> ( C =/= (/) -> 0 <_ sup ( C , RR , < ) ) )
96	51, 95	mpd 15	. . . . . . . . . . 11 |- ( ph -> 0 <_ sup ( C , RR , < ) )
97	96	anim1i 578	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
98		0red 9662	. . . . . . . . . . . 12 |- ( ph -> 0 e. RR )
99		lelttr 9742	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ sup ( C , RR , < ) e. RR /\ ( A x. sup ( B , RR , < ) ) e. RR ) -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
100	98, 63, 52, 99	syl3anc 1292	. . . . . . . . . . 11 |- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
101	100	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
102	97, 101	mpd 15	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) )
103		prodgt02 10473	. . . . . . . . 9 |- ( ( ( A e. RR /\ sup ( B , RR , < ) e. RR ) /\ ( 0 <_ sup ( B , RR , < ) /\ 0 < ( A x. sup ( B , RR , < ) ) ) ) -> 0 < A )
104	66, 65, 79, 102, 103	syl22anc 1293	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A )
105		ltdivmul 10502	. . . . . . . 8 |- ( ( sup ( C , RR , < ) e. RR /\ sup ( B , RR , < ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
106	64, 65, 66, 104, 105	syl112anc 1296	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
107	61, 106	mpbird 240	. . . . . 6 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) )
108	12	adantr 472	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
109	104	gt0ne0d 10199	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 )
110	64, 66, 109	redivcld 10457	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR )
111		suprlub 10593	. . . . . . 7 |- ( ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) /\ ( sup ( C , RR , < ) / A ) e. RR ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> E. b e. B ( sup ( C , RR , < ) / A ) < b ) )
112	108, 110, 111	syl2anc 673	. . . . . 6 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> E. b e. B ( sup ( C , RR , < ) / A ) < b ) )
113	107, 112	mpbid 215	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> E. b e. B ( sup ( C , RR , < ) / A ) < b )
114		rspe 2844	. . . . . . . . . . . . . . 15 |- ( ( b e. B /\ w = ( A x. b ) ) -> E. b e. B w = ( A x. b ) )
115	114, 9	sylibr 217	. . . . . . . . . . . . . 14 |- ( ( b e. B /\ w = ( A x. b ) ) -> w e. C )
116	115	adantl 473	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C )
117		simplrr 779	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) )
118	91	adantlr 729	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
119	117, 118	eqbrtrrd 4418	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) )
120	116, 119	mpdan 681	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) )
121	120	expr 626	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
122	121	exlimdv 1787	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( E. w w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
123	43, 122	mpi 20	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
124	123	adantlr 729	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
125	34	adantlr 729	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR )
126	63	ad2antrr 740	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR )
127	125, 126	lenltd 9798	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( ( A x. b ) <_ sup ( C , RR , < ) <-> -. sup ( C , RR , < ) < ( A x. b ) ) )
128	124, 127	mpbid 215	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. sup ( C , RR , < ) < ( A x. b ) )
129	14	adantlr 729	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR )
130	19	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR )
131	104	adantr 472	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A )
132		ltdivmul 10502	. . . . . . . 8 |- ( ( sup ( C , RR , < ) e. RR /\ b e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( sup ( C , RR , < ) / A ) < b <-> sup ( C , RR , < ) < ( A x. b ) ) )
133	126, 129, 130, 131, 132	syl112anc 1296	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( ( sup ( C , RR , < ) / A ) < b <-> sup ( C , RR , < ) < ( A x. b ) ) )
134	128, 133	mtbird 308	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. ( sup ( C , RR , < ) / A ) < b )
135	134	nrexdv 2842	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b )
136	113, 135	pm2.65da 586	. . . 4 |- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
137	60, 136	jca 541	. . 3 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
138	63, 52	eqleltd 9796	. . 3 |- ( ph -> ( sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) <-> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) )
139	137, 138	mpbird 240	. 2 |- ( ph -> sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) )
140	139	eqcomd 2477	1 |- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   {cab 2457   =/= wne 2641   A.wral 2756   E.wrex 2757   C_ wss 3390   (/)c0 3722   class class class wbr 4395  (class class class)co 6308   supcsup 7972   RRcr 9556   0cc0 9557   x. cmul 9562   < clt 9693   <_ cle 9694   / cdiv 10291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292

This theorem is referenced by:  supmul  10601  hoidmvlelem1  38535