Theorem supmul1 9929

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 9932 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 2919	. . . . . . . 8 |- w e. _V
2		oveq2 6048	. . . . . . . . . . 11 |- ( v = b -> ( A x. v ) = ( A x. b ) )
3	2	eqeq2d 2415	. . . . . . . . . 10 |- ( v = b -> ( z = ( A x. v ) <-> z = ( A x. b ) ) )
4	3	cbvrexv 2893	. . . . . . . . 9 |- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) )
5		eqeq1 2410	. . . . . . . . . 10 |- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) )
6	5	rexbidv 2687	. . . . . . . . 9 |- ( z = w -> ( E. b e. B z = ( A x. b ) <-> E. b e. B w = ( A x. b ) ) )
7	4, 6	syl5bb 249	. . . . . . . 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 3045	. . . . . . 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 448	. . . . . . . . . . . . 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 188	. . . . . . . . . . . 12 |- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
13	12	simp1d 969	. . . . . . . . . . 11 |- ( ph -> B C_ RR )
14	13	sselda 3308	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b e. RR )
15		suprcl 9924	. . . . . . . . . . . 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 452	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR )
18		simpl1 960	. . . . . . . . . . . . 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 188	. . . . . . . . . . . 12 |- ( ph -> A e. RR )
20		simpl2 961	. . . . . . . . . . . . 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 188	. . . . . . . . . . . 12 |- ( ph -> 0 <_ A )
22	19, 21	jca 519	. . . . . . . . . . 11 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
23	22	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) )
24		suprub 9925	. . . . . . . . . . 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 458	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) )
26		lemul2a 9821	. . . . . . . . . 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 1187	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) )
28		breq1 4175	. . . . . . . . 9 |- ( w = ( A x. b ) -> ( w <_ ( A x. sup ( B , RR , < ) ) <-> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) )
29	27, 28	syl5ibrcom 214	. . . . . . . 8 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
30	29	rexlimdva 2790	. . . . . . 7 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) )
31	9, 30	syl5bi 209	. . . . . 6 |- ( ph -> ( w e. C -> w <_ ( A x. sup ( B , RR , < ) ) ) )
32	31	ralrimiv 2748	. . . . 5 |- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) )
33	19	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ b e. B ) -> A e. RR )
34	33, 14	remulcld 9072	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR )
35		eleq1a 2473	. . . . . . . . . . 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 2790	. . . . . . . . 9 |- ( ph -> ( E. b e. B w = ( A x. b ) -> w e. RR ) )
38	9, 37	syl5bi 209	. . . . . . . 8 |- ( ph -> ( w e. C -> w e. RR ) )
39	38	ssrdv 3314	. . . . . . 7 |- ( ph -> C C_ RR )
40		simpr2 964	. . . . . . . . . 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 188	. . . . . . . . 9 |- ( ph -> B =/= (/) )
42		ovex 6065	. . . . . . . . . . 11 |- ( A x. b ) e. _V
43	42	isseti 2922	. . . . . . . . . 10 |- E. w w = ( A x. b )
44	43	rgenw 2733	. . . . . . . . 9 |- A. b e. B E. w w = ( A x. b )
45		r19.2z 3677	. . . . . . . . 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 644	. . . . . . . 8 |- ( ph -> E. b e. B E. w w = ( A x. b ) )
47	9	exbii 1589	. . . . . . . . 9 |- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) )
48		n0 3597	. . . . . . . . 9 |- ( C =/= (/) <-> E. w w e. C )
49		rexcom4 2935	. . . . . . . . 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 269	. . . . . . . 8 |- ( C =/= (/) <-> E. b e. B E. w w = ( A x. b ) )
51	46, 50	sylibr 204	. . . . . . 7 |- ( ph -> C =/= (/) )
52	19, 16	remulcld 9072	. . . . . . . 8 |- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR )
53		breq2 4176	. . . . . . . . . 10 |- ( x = ( A x. sup ( B , RR , < ) ) -> ( w <_ x <-> w <_ ( A x. sup ( B , RR , < ) ) ) )
54	53	ralbidv 2686	. . . . . . . . 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 3012	. . . . . . . 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 643	. . . . . . 7 |- ( ph -> E. x e. RR A. w e. C w <_ x )
57	39, 51, 56	3jca 1134	. . . . . 6 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
58		suprleub 9928	. . . . . 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 643	. . . . 5 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) )
60	32, 59	mpbird 224	. . . 4 |- ( ph -> sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) )
61		simpr 448	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
62		suprcl 9924	. . . . . . . . . 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 452	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR )
65	16	adantr 452	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR )
66	19	adantr 452	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR )
67		n0 3597	. . . . . . . . . . . 12 |- ( B =/= (/) <-> E. b b e. B )
68		0re 9047	. . . . . . . . . . . . . . . 16 |- 0 e. RR
69	68	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 e. RR )
70		simpl3 962	. . . . . . . . . . . . . . . . 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 )
71	10, 70	sylbi 188	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. B 0 <_ x )
72		breq2 4176	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ( 0 <_ x <-> 0 <_ b ) )
73	72	rspccva 3011	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b )
74	71, 73	sylan 458	. . . . . . . . . . . . . . 15 |- ( ( ph /\ b e. B ) -> 0 <_ b )
75	69, 14, 17, 74, 25	letrd 9183	. . . . . . . . . . . . . 14 |- ( ( ph /\ b e. B ) -> 0 <_ sup ( B , RR , < ) )
76	75	ex 424	. . . . . . . . . . . . 13 |- ( ph -> ( b e. B -> 0 <_ sup ( B , RR , < ) ) )
77	76	exlimdv 1643	. . . . . . . . . . . 12 |- ( ph -> ( E. b b e. B -> 0 <_ sup ( B , RR , < ) ) )
78	67, 77	syl5bi 209	. . . . . . . . . . 11 |- ( ph -> ( B =/= (/) -> 0 <_ sup ( B , RR , < ) ) )
79	41, 78	mpd 15	. . . . . . . . . 10 |- ( ph -> 0 <_ sup ( B , RR , < ) )
80	79	adantr 452	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) )
81	68	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 e. RR )
82	38	imp 419	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w e. RR )
83	63	adantr 452	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR )
84	21	adantr 452	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ b e. B ) -> 0 <_ A )
85	33, 14, 84, 74	mulge0d 9559	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) )
86		breq2 4176	. . . . . . . . . . . . . . . . . . . 20 |- ( w = ( A x. b ) -> ( 0 <_ w <-> 0 <_ ( A x. b ) ) )
87	85, 86	syl5ibrcom 214	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> 0 <_ w ) )
88	87	rexlimdva 2790	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( E. b e. B w = ( A x. b ) -> 0 <_ w ) )
89	9, 88	syl5bi 209	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( w e. C -> 0 <_ w ) )
90	89	imp 419	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> 0 <_ w )
91		suprub 9925	. . . . . . . . . . . . . . . . 17 |- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
92	57, 91	sylan 458	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
93	81, 82, 83, 90, 92	letrd 9183	. . . . . . . . . . . . . . 15 |- ( ( ph /\ w e. C ) -> 0 <_ sup ( C , RR , < ) )
94	93	ex 424	. . . . . . . . . . . . . 14 |- ( ph -> ( w e. C -> 0 <_ sup ( C , RR , < ) ) )
95	94	exlimdv 1643	. . . . . . . . . . . . 13 |- ( ph -> ( E. w w e. C -> 0 <_ sup ( C , RR , < ) ) )
96	48, 95	syl5bi 209	. . . . . . . . . . . 12 |- ( ph -> ( C =/= (/) -> 0 <_ sup ( C , RR , < ) ) )
97	51, 96	mpd 15	. . . . . . . . . . 11 |- ( ph -> 0 <_ sup ( C , RR , < ) )
98	97	anim1i 552	. . . . . . . . . 10 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
99	68	a1i 11	. . . . . . . . . . . 12 |- ( ph -> 0 e. RR )
100		lelttr 9121	. . . . . . . . . . . 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 , < ) ) ) )
101	99, 63, 52, 100	syl3anc 1184	. . . . . . . . . . 11 |- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) )
102	101	adantr 452	. . . . . . . . . 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 , < ) ) ) )
103	98, 102	mpd 15	. . . . . . . . 9 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) )
104		prodgt02 9812	. . . . . . . . 9 |- ( ( ( A e. RR /\ sup ( B , RR , < ) e. RR ) /\ ( 0 <_ sup ( B , RR , < ) /\ 0 < ( A x. sup ( B , RR , < ) ) ) ) -> 0 < A )
105	66, 65, 80, 103, 104	syl22anc 1185	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A )
106		ltdivmul 9838	. . . . . . . 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 , < ) ) ) )
107	64, 65, 66, 105, 106	syl112anc 1188	. . . . . . 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 , < ) ) ) )
108	61, 107	mpbird 224	. . . . . 6 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) )
109	12	adantr 452	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) )
110	105	gt0ne0d 9547	. . . . . . . 8 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 )
111	64, 66, 110	redivcld 9798	. . . . . . 7 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR )
112		suprlub 9926	. . . . . . 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 ) )
113	109, 111, 112	syl2anc 643	. . . . . 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 ) )
114	108, 113	mpbid 202	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> E. b e. B ( sup ( C , RR , < ) / A ) < b )
115		rspe 2727	. . . . . . . . . . . . . . 15 |- ( ( b e. B /\ w = ( A x. b ) ) -> E. b e. B w = ( A x. b ) )
116	115, 9	sylibr 204	. . . . . . . . . . . . . 14 |- ( ( b e. B /\ w = ( A x. b ) ) -> w e. C )
117	116	adantl 453	. . . . . . . . . . . . 13 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C )
118		simplrr 738	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) )
119	92	adantlr 696	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
120	118, 119	eqbrtrrd 4194	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) )
121	117, 120	mpdan 650	. . . . . . . . . . . 12 |- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) )
122	121	expr 599	. . . . . . . . . . 11 |- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
123	122	exlimdv 1643	. . . . . . . . . 10 |- ( ( ph /\ b e. B ) -> ( E. w w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) )
124	43, 123	mpi 17	. . . . . . . . 9 |- ( ( ph /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
125	124	adantlr 696	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) )
126	34	adantlr 696	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR )
127	63	ad2antrr 707	. . . . . . . . 9 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR )
128	126, 127	lenltd 9175	. . . . . . . 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 ) ) )
129	125, 128	mpbid 202	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. sup ( C , RR , < ) < ( A x. b ) )
130	14	adantlr 696	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR )
131	19	ad2antrr 707	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR )
132	105	adantr 452	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A )
133		ltdivmul 9838	. . . . . . . 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 ) ) )
134	127, 130, 131, 132, 133	syl112anc 1188	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( ( sup ( C , RR , < ) / A ) < b <-> sup ( C , RR , < ) < ( A x. b ) ) )
135	129, 134	mtbird 293	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. ( sup ( C , RR , < ) / A ) < b )
136	135	nrexdv 2769	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b )
137	114, 136	pm2.65da 560	. . . 4 |- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) )
138	60, 137	jca 519	. . 3 |- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) )
139	63, 52	eqleltd 9173	. . 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 , < ) ) ) ) )
140	138, 139	mpbird 224	. 2 |- ( ph -> sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) )
141	140	eqcomd 2409	1 |- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   {cab 2390   =/= wne 2567   A.wral 2666   E.wrex 2667   C_ wss 3280   (/)c0 3588   class class class wbr 4172  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946   x. cmul 8951   < clt 9076   <_ cle 9077   / cdiv 9633

This theorem is referenced by:  supmul  9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634