Theorem supmul1 10291

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 10294 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 2973	. . . . . . . 8 |- w e. _V
2		oveq2 6098	. . . . . . . . . . 11 |- ( v = b -> ( A x. v ) = ( A x. b ) )
3	2	eqeq2d 2452	. . . . . . . . . 10 |- ( v = b -> ( z = ( A x. v ) <-> z = ( A x. b ) ) )
4	3	cbvrexv 2946	. . . . . . . . 9 |- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) )
5		eqeq1 2447	. . . . . . . . . 10 |- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) )
6	5	rexbidv 2734	. . . . . . . . 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 3106	. . . . . . 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 458	. . . . . . . . . . . . 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 995	. . . . . . . . . . 11 |- ( ph -> B C_ RR )
14	13	sselda 3353	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b e. RR )
15		suprcl 10286	. . . . . . . . . . . 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 462	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR )
18		simpl1 986	. . . . . . . . . . . . 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 987	. . . . . . . . . . . . 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 529	. . . . . . . . . . 11 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
23	22	adantr 462	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) )
24		suprub 10287	. . . . . . . . . . 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 468	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) )
26		lemul2a 10180	. . . . . . . . . 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 1216	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
28		breq1 4292	. . . . . . . . 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 2839	. . . . . . 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 2796	. . . . 5 |- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) )
33	19	adantr 462	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B ) -> A e. RR )
34	33, 14	remulcld 9410	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR )
35		eleq1a 2510	. . . . . . . . . . 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 2839	. . . . . . . . 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 3359	. . . . . . 7 |- ( ph -> C C_ RR )
40		simpr2 990	. . . . . . . . . 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 6115	. . . . . . . . . . 11 |- ( A x. b ) e. _V
43	42	isseti 2976	. . . . . . . . . 10 |- E. w w = ( A x. b )
44	43	rgenw 2781	. . . . . . . . 9 |- A. b e. B E. w w = ( A x. b )
45		r19.2z 3766	. . . . . . . . 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 657	. . . . . . . 8 |- ( ph -> E. b e. B E. w w = ( A x. b ) )
47	9	exbii 1639	. . . . . . . . 9 |- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) )
48		n0 3643	. . . . . . . . 9 |- ( C =/= (/) <-> E. w w e. C )
49		rexcom4 2990	. . . . . . . . 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 9410	. . . . . . . 8 |- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR )
53		breq2 4293	. . . . . . . . . 10 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( A x. sup ( B , RR , < ) ) ) )
54	53	ralbidv 2733	. . . . . . . . 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 3070	. . . . . . . 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 656	. . . . . . 7 |- ( ph -> E. x e. RR A. w e. C w <_ x )
57	39, 51, 56	3jca 1163	. . . . . 6 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
58		suprleub 10290	. . . . . 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 656	. . . . 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 458	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
62		suprcl 10286	. . . . . . . . . 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 462	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR )
65	16	adantr 462	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR )
66	19	adantr 462	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR )
67		n0 3643	. . . . . . . . . . . 12 |- ( B =/= (/) <-> E. b b e. B )
68		0red 9383	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 e. RR )
69		simpl3 988	. . . . . . . . . . . . . . . . 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 4293	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( 0 <_ x <-> 0 <_ b ) )
72	71	rspccva 3069	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b )
73	70, 72	sylan 468	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 <_ b )
74	68, 14, 17, 73, 25	letrd 9524	. . . . . . . . . . . . . 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 1695	. . . . . . . . . . . 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 462	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) )
80		0red 9383	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 e. RR )
81	38	imp 429	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w e. RR )
82	63	adantr 462	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR )
83	21	adantr 462	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ b e. B ) -> 0 <_ A )
84	33, 14, 83, 73	mulge0d 9912	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) )
85		breq2 4293	. . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . 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 10287	. . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
92	80, 81, 82, 89, 91	letrd 9524	. . . . . . . . . . . . . . 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 1695	. . . . . . . . . . . . 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 565	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
98		0red 9383	. . . . . . . . . . . 12 |- ( ph -> 0 e. RR )
99		lelttr 9461	. . . . . . . . . . . 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 1213	. . . . . . . . . . 11 |- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
101	100	adantr 462	. . . . . . . . . 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 10171	. . . . . . . . 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 1214	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A )
105		ltdivmul 10200	. . . . . . . 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 1217	. . . . . . 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 462	. . . . . . 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 9900	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 )
110	64, 66, 109	redivcld 10155	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR )
111		suprlub 10288	. . . . . . 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 656	. . . . . 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 2775	. . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C )
117		simplrr 755	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) )
118	91	adantlr 709	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
119	117, 118	eqbrtrrd 4311	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) )
120	116, 119	mpdan 663	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) )
121	120	expr 612	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
122	121	exlimdv 1695	. . . . . . . . . 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 709	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
125	34	adantlr 709	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR )
126	63	ad2antrr 720	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR )
127	125, 126	lenltd 9516	. . . . . . . 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 709	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR )
130	19	ad2antrr 720	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR )
131	104	adantr 462	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A )
132		ltdivmul 10200	. . . . . . . 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 1217	. . . . . . 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 2817	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b )
136	113, 135	pm2.65da 573	. . . 4 |- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
137	60, 136	jca 529	. . 3 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
138	63, 52	eqleltd 9514	. . 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 2446	1 |- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   E.wex 1591   e. wcel 1761   {cab 2427   =/= wne 2604   A.wral 2713   E.wrex 2714   C_ wss 3325   (/)c0 3634   class class class wbr 4289  (class class class)co 6090   supcsup 7686   RRcr 9277   0cc0 9278   x. cmul 9283   < clt 9414   <_ cle 9415   / cdiv 9989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990

This theorem is referenced by:  supmul  10294