Theorem supmul1 10507

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 10510 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 3116	. . . . . . . 8 |- w e. _V
2		oveq2 6291	. . . . . . . . . . 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 3089	. . . . . . . . 9 |- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) )
5		eqeq1 2471	. . . . . . . . . 10 |- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) )
6	5	rexbidv 2973	. . . . . . . . 9 |- ( z = w -> ( E. b e. B z = ( A x. b ) <-> E. b e. B w = ( A x. b ) ) )
7	4, 6	syl5bb 257	. . . . . . . 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 3253	. . . . . . 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 461	. . . . . . . . . . . . 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 195	. . . . . . . . . . . 12 |- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
13	12	simp1d 1008	. . . . . . . . . . 11 |- ( ph -> B C_ RR )
14	13	sselda 3504	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b e. RR )
15		suprcl 10502	. . . . . . . . . . . 12 |- ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) -> sup ( B , RR , < ) e. RR )
16	12, 15	syl 16	. . . . . . . . . . 11 |- ( ph -> sup ( B , RR , < ) e. RR )
17	16	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR )
18		simpl1 999	. . . . . . . . . . . . 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 195	. . . . . . . . . . . 12 |- ( ph -> A e. RR )
20		simpl2 1000	. . . . . . . . . . . . 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 195	. . . . . . . . . . . 12 |- ( ph -> 0 <_ A )
22	19, 21	jca 532	. . . . . . . . . . 11 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
23	22	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) )
24		suprub 10503	. . . . . . . . . . 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 471	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) )
26		lemul2a 10396	. . . . . . . . . 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 1231	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
28		breq1 4450	. . . . . . . . 9 |- ( w = ( A x. b ) -> ( w <_ ( A x. sup ( B , RR , < ) ) <-> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) )
29	27, 28	syl5ibrcom 222	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
30	29	rexlimdva 2955	. . . . . . 7 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
31	9, 30	syl5bi 217	. . . . . 6 |- ( ph -> ( w e. C -> w <_ ( A x. sup ( B , RR , < ) ) ) )
32	31	ralrimiv 2876	. . . . 5 |- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) )
33	19	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B ) -> A e. RR )
34	33, 14	remulcld 9623	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR )
35		eleq1a 2550	. . . . . . . . . . 11 |- ( ( A x. b ) e. RR -> ( w = ( A x. b ) -> w e. RR ) )
36	34, 35	syl 16	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w e. RR ) )
37	36	rexlimdva 2955	. . . . . . . . 9 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w e. RR ) )
38	9, 37	syl5bi 217	. . . . . . . 8 |- ( ph -> ( w e. C -> w e. RR ) )
39	38	ssrdv 3510	. . . . . . 7 |- ( ph -> C C_ RR )
40		simpr2 1003	. . . . . . . . . 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 195	. . . . . . . . 9 |- ( ph -> B =/= (/) )
42		ovex 6308	. . . . . . . . . . 11 |- ( A x. b ) e. _V
43	42	isseti 3119	. . . . . . . . . 10 |- E. w w = ( A x. b )
44	43	rgenw 2825	. . . . . . . . 9 |- A. b e. B E. w w = ( A x. b )
45		r19.2z 3917	. . . . . . . . 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 662	. . . . . . . 8 |- ( ph -> E. b e. B E. w w = ( A x. b ) )
47	9	exbii 1644	. . . . . . . . 9 |- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) )
48		n0 3794	. . . . . . . . 9 |- ( C =/= (/) <-> E. w w e. C )
49		rexcom4 3133	. . . . . . . . 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 277	. . . . . . . 8 |- ( C =/= (/) <-> E. b e. B E. w w = ( A x. b ) )
51	46, 50	sylibr 212	. . . . . . 7 |- ( ph -> C =/= (/) )
52	19, 16	remulcld 9623	. . . . . . . 8 |- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR )
53		breq2 4451	. . . . . . . . . 10 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( A x. sup ( B , RR , < ) ) ) )
54	53	ralbidv 2903	. . . . . . . . 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 3214	. . . . . . . 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 661	. . . . . . 7 |- ( ph -> E. x e. RR A. w e. C w <_ x )
57	39, 51, 56	3jca 1176	. . . . . 6 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
58		suprleub 10506	. . . . . 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 661	. . . . 5 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
60	32, 59	mpbird 232	. . . 4 |- ( ph -> sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) )
61		simpr 461	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
62		suprcl 10502	. . . . . . . . . 10 |- ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) -> sup ( C , RR , < ) e. RR )
63	57, 62	syl 16	. . . . . . . . 9 |- ( ph -> sup ( C , RR , < ) e. RR )
64	63	adantr 465	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR )
65	16	adantr 465	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR )
66	19	adantr 465	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR )
67		n0 3794	. . . . . . . . . . . 12 |- ( B =/= (/) <-> E. b b e. B )
68		0red 9596	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 e. RR )
69		simpl3 1001	. . . . . . . . . . . . . . . . 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 195	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. B 0 <_ x )
71		breq2 4451	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( 0 <_ x <-> 0 <_ b ) )
72	71	rspccva 3213	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b )
73	70, 72	sylan 471	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 <_ b )
74	68, 14, 17, 73, 25	letrd 9737	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> 0 <_ sup ( B , RR , < ) )
75	74	ex 434	. . . . . . . . . . . . 13 |- ( ph -> ( b e. B -> 0 <_ sup ( B , RR , < ) ) )
76	75	exlimdv 1700	. . . . . . . . . . . 12 |- ( ph -> ( E. b b e. B -> 0 <_ sup ( B , RR , < ) ) )
77	67, 76	syl5bi 217	. . . . . . . . . . 11 |- ( ph -> ( B =/= (/) -> 0 <_ sup ( B , RR , < ) ) )
78	41, 77	mpd 15	. . . . . . . . . 10 |- ( ph -> 0 <_ sup ( B , RR , < ) )
79	78	adantr 465	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) )
80		0red 9596	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 e. RR )
81	38	imp 429	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w e. RR )
82	63	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR )
83	21	adantr 465	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ b e. B ) -> 0 <_ A )
84	33, 14, 83, 73	mulge0d 10128	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) )
85		breq2 4451	. . . . . . . . . . . . . . . . . . . 20 |- ( w = ( A x. b ) -> ( 0 <_ w <-> 0 <_ ( A x. b ) ) )
86	84, 85	syl5ibrcom 222	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> 0 <_ w ) )
87	86	rexlimdva 2955	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( E. b e. B w = ( A x. b ) -> 0 <_ w ) )
88	9, 87	syl5bi 217	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( w e. C -> 0 <_ w ) )
89	88	imp 429	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 <_ w )
90		suprub 10503	. . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
92	80, 81, 82, 89, 91	letrd 9737	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. C ) -> 0 <_ sup ( C , RR , < ) )
93	92	ex 434	. . . . . . . . . . . . . 14 |- ( ph -> ( w e. C -> 0 <_ sup ( C , RR , < ) ) )
94	93	exlimdv 1700	. . . . . . . . . . . . 13 |- ( ph -> ( E. w w e. C -> 0 <_ sup ( C , RR , < ) ) )
95	48, 94	syl5bi 217	. . . . . . . . . . . 12 |- ( ph -> ( C =/= (/) -> 0 <_ sup ( C , RR , < ) ) )
96	51, 95	mpd 15	. . . . . . . . . . 11 |- ( ph -> 0 <_ sup ( C , RR , < ) )
97	96	anim1i 568	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
98		0red 9596	. . . . . . . . . . . 12 |- ( ph -> 0 e. RR )
99		lelttr 9674	. . . . . . . . . . . 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 1228	. . . . . . . . . . 11 |- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
101	100	adantr 465	. . . . . . . . . 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 10387	. . . . . . . . 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 1229	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A )
105		ltdivmul 10416	. . . . . . . 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 1232	. . . . . . 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 232	. . . . . 6 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) )
108	12	adantr 465	. . . . . . 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 10116	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 )
110	64, 66, 109	redivcld 10371	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR )
111		suprlub 10504	. . . . . . 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 661	. . . . . 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 210	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> E. b e. B ( sup ( C , RR , < ) / A ) < b )
114		rspe 2922	. . . . . . . . . . . . . . 15 |- ( ( b e. B /\ w = ( A x. b ) ) -> E. b e. B w = ( A x. b ) )
115	114, 9	sylibr 212	. . . . . . . . . . . . . 14 |- ( ( b e. B /\ w = ( A x. b ) ) -> w e. C )
116	115	adantl 466	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C )
117		simplrr 760	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) )
118	91	adantlr 714	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
119	117, 118	eqbrtrrd 4469	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) )
120	116, 119	mpdan 668	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) )
121	120	expr 615	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
122	121	exlimdv 1700	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( E. w w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
123	43, 122	mpi 17	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
124	123	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
125	34	adantlr 714	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR )
126	63	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR )
127	125, 126	lenltd 9729	. . . . . . . 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 210	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. sup ( C , RR , < ) < ( A x. b ) )
129	14	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR )
130	19	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR )
131	104	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A )
132		ltdivmul 10416	. . . . . . . 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 1232	. . . . . . 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 301	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. ( sup ( C , RR , < ) / A ) < b )
135	134	nrexdv 2920	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b )
136	113, 135	pm2.65da 576	. . . 4 |- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
137	60, 136	jca 532	. . 3 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
138	63, 52	eqleltd 9727	. . 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 232	. 2 |- ( ph -> sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) )
140	139	eqcomd 2475	1 |- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   E.wrex 2815   C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6283   supcsup 7899   RRcr 9490   0cc0 9491   x. cmul 9496   < clt 9627   <_ cle 9628   / cdiv 10205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206

This theorem is referenced by:  supmul  10510