Theorem supmul1 10992

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 10995 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 3203	. . . . . . . 8 |- w e. _V
2		oveq2 6658	. . . . . . . . . . 11 |- ( v = b -> ( A x. v ) = ( A x. b ) )
3	2	eqeq2d 2632	. . . . . . . . . 10 |- ( v = b -> ( z = ( A x. v ) <-> z = ( A x. b ) ) )
4	3	cbvrexv 3172	. . . . . . . . 9 |- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) )
5		eqeq1 2626	. . . . . . . . . 10 |- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) )
6	5	rexbidv 3052	. . . . . . . . 9 |- ( z = w -> ( E. b e. B z = ( A x. b ) <-> E. b e. B w = ( A x. b ) ) )
7	4, 6	syl5bb 272	. . . . . . . 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 3354	. . . . . . 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 477	. . . . . . . . . . . . 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 207	. . . . . . . . . . . 12 |- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
13	12	simp1d 1073	. . . . . . . . . . 11 |- ( ph -> B C_ RR )
14	13	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b e. RR )
15		suprcl 10983	. . . . . . . . . . . 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 481	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR )
18		simpl1 1064	. . . . . . . . . . . . 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 207	. . . . . . . . . . . 12 |- ( ph -> A e. RR )
20		simpl2 1065	. . . . . . . . . . . . 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 207	. . . . . . . . . . . 12 |- ( ph -> 0 <_ A )
22	19, 21	jca 554	. . . . . . . . . . 11 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
23	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) )
24		suprub 10984	. . . . . . . . . . 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 488	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) )
26		lemul2a 10878	. . . . . . . . . 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 1329	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
28		breq1 4656	. . . . . . . . 9 |- ( w = ( A x. b ) -> ( w <_ ( A x. sup ( B , RR , < ) ) <-> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) )
29	27, 28	syl5ibrcom 237	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
30	29	rexlimdva 3031	. . . . . . 7 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
31	9, 30	syl5bi 232	. . . . . 6 |- ( ph -> ( w e. C -> w <_ ( A x. sup ( B , RR , < ) ) ) )
32	31	ralrimiv 2965	. . . . 5 |- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) )
33	19	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B ) -> A e. RR )
34	33, 14	remulcld 10070	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR )
35		eleq1a 2696	. . . . . . . . . . 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 3031	. . . . . . . . 9 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w e. RR ) )
38	9, 37	syl5bi 232	. . . . . . . 8 |- ( ph -> ( w e. C -> w e. RR ) )
39	38	ssrdv 3609	. . . . . . 7 |- ( ph -> C C_ RR )
40		simpr2 1068	. . . . . . . . . 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 207	. . . . . . . . 9 |- ( ph -> B =/= (/) )
42		ovex 6678	. . . . . . . . . . 11 |- ( A x. b ) e. _V
43	42	isseti 3209	. . . . . . . . . 10 |- E. w w = ( A x. b )
44	43	rgenw 2924	. . . . . . . . 9 |- A. b e. B E. w w = ( A x. b )
45		r19.2z 4060	. . . . . . . . 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 694	. . . . . . . 8 |- ( ph -> E. b e. B E. w w = ( A x. b ) )
47	9	exbii 1774	. . . . . . . . 9 |- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) )
48		n0 3931	. . . . . . . . 9 |- ( C =/= (/) <-> E. w w e. C )
49		rexcom4 3225	. . . . . . . . 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 292	. . . . . . . 8 |- ( C =/= (/) <-> E. b e. B E. w w = ( A x. b ) )
51	46, 50	sylibr 224	. . . . . . 7 |- ( ph -> C =/= (/) )
52	19, 16	remulcld 10070	. . . . . . . 8 |- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR )
53		breq2 4657	. . . . . . . . . 10 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( A x. sup ( B , RR , < ) ) ) )
54	53	ralbidv 2986	. . . . . . . . 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 3309	. . . . . . . 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 693	. . . . . . 7 |- ( ph -> E. x e. RR A. w e. C w <_ x )
57	39, 51, 56	3jca 1242	. . . . . 6 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
58		suprleub 10989	. . . . . 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 693	. . . . 5 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
60	32, 59	mpbird 247	. . . 4 |- ( ph -> sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) )
61		simpr 477	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
62		suprcl 10983	. . . . . . . . . 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 481	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR )
65	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR )
66	19	adantr 481	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR )
67		n0 3931	. . . . . . . . . . . 12 |- ( B =/= (/) <-> E. b b e. B )
68		0red 10041	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 e. RR )
69		simpl3 1066	. . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. B 0 <_ x )
71		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( 0 <_ x <-> 0 <_ b ) )
72	71	rspccva 3308	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b )
73	70, 72	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 <_ b )
74	68, 14, 17, 73, 25	letrd 10194	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> 0 <_ sup ( B , RR , < ) )
75	74	ex 450	. . . . . . . . . . . . 13 |- ( ph -> ( b e. B -> 0 <_ sup ( B , RR , < ) ) )
76	75	exlimdv 1861	. . . . . . . . . . . 12 |- ( ph -> ( E. b b e. B -> 0 <_ sup ( B , RR , < ) ) )
77	67, 76	syl5bi 232	. . . . . . . . . . 11 |- ( ph -> ( B =/= (/) -> 0 <_ sup ( B , RR , < ) ) )
78	41, 77	mpd 15	. . . . . . . . . 10 |- ( ph -> 0 <_ sup ( B , RR , < ) )
79	78	adantr 481	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) )
80		0red 10041	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 e. RR )
81	38	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w e. RR )
82	63	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR )
83	21	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ b e. B ) -> 0 <_ A )
84	33, 14, 83, 73	mulge0d 10604	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) )
85		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 |- ( w = ( A x. b ) -> ( 0 <_ w <-> 0 <_ ( A x. b ) ) )
86	84, 85	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> 0 <_ w ) )
87	86	rexlimdva 3031	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( E. b e. B w = ( A x. b ) -> 0 <_ w ) )
88	9, 87	syl5bi 232	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( w e. C -> 0 <_ w ) )
89	88	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 <_ w )
90		suprub 10984	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
92	80, 81, 82, 89, 91	letrd 10194	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. C ) -> 0 <_ sup ( C , RR , < ) )
93	92	ex 450	. . . . . . . . . . . . . 14 |- ( ph -> ( w e. C -> 0 <_ sup ( C , RR , < ) ) )
94	93	exlimdv 1861	. . . . . . . . . . . . 13 |- ( ph -> ( E. w w e. C -> 0 <_ sup ( C , RR , < ) ) )
95	48, 94	syl5bi 232	. . . . . . . . . . . 12 |- ( ph -> ( C =/= (/) -> 0 <_ sup ( C , RR , < ) ) )
96	51, 95	mpd 15	. . . . . . . . . . 11 |- ( ph -> 0 <_ sup ( C , RR , < ) )
97	96	anim1i 592	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
98		0red 10041	. . . . . . . . . . . 12 |- ( ph -> 0 e. RR )
99		lelttr 10128	. . . . . . . . . . . 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 1326	. . . . . . . . . . 11 |- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
101	100	adantr 481	. . . . . . . . . 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 10869	. . . . . . . . 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 1327	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A )
105		ltdivmul 10898	. . . . . . . 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 1330	. . . . . . 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 247	. . . . . 6 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) )
108	12	adantr 481	. . . . . . 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 10592	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 )
110	64, 66, 109	redivcld 10853	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR )
111		suprlub 10987	. . . . . . 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 693	. . . . . 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 222	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> E. b e. B ( sup ( C , RR , < ) / A ) < b )
114		rspe 3003	. . . . . . . . . . . . . . 15 |- ( ( b e. B /\ w = ( A x. b ) ) -> E. b e. B w = ( A x. b ) )
115	114, 9	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( b e. B /\ w = ( A x. b ) ) -> w e. C )
116	115	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C )
117		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) )
118	91	adantlr 751	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
119	117, 118	eqbrtrrd 4677	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) )
120	116, 119	mpdan 702	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) )
121	120	expr 643	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
122	121	exlimdv 1861	. . . . . . . . . 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 751	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
125	34	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR )
126	63	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR )
127	125, 126	lenltd 10183	. . . . . . . 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 222	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. sup ( C , RR , < ) < ( A x. b ) )
129	14	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR )
130	19	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR )
131	104	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A )
132		ltdivmul 10898	. . . . . . . 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 1330	. . . . . . 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 315	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. ( sup ( C , RR , < ) / A ) < b )
135	134	nrexdv 3001	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b )
136	113, 135	pm2.65da 600	. . . 4 |- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
137	60, 136	jca 554	. . 3 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
138	63, 52	eqleltd 10181	. . 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 247	. 2 |- ( ph -> sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) )
140	139	eqcomd 2628	1 |- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684

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-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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685

This theorem is referenced by:  supmul  10995  hoidmvlelem1  40809