Theorem ivthlem1 23220

Description: Lemma for ivth 23223. The set S of all x values with ( F ` x ) less than U is lower bounded by A and upper bounded by B. (Contributed by Mario Carneiro, 17-Jun-2014.)

Hypotheses

Ref	Expression
ivth.1	|- ( ph -> A e. RR )
ivth.2	|- ( ph -> B e. RR )
ivth.3	|- ( ph -> U e. RR )
ivth.4	|- ( ph -> A < B )
ivth.5	|- ( ph -> ( A [,] B ) C_ D )
ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivth.10	|- S = { x e. ( A [,] B ) | ( F ` x ) <_ U }

Assertion

Ref	Expression
ivthlem1	|- ( ph -> ( A e. S /\ A. z e. S z <_ B ) )

Distinct variable groups:   x, z, B   x, D, z   x, F, z   ph, x, z   x, A   x, S, z   x, U, z

Allowed substitution hint:   A( z)

Proof of Theorem ivthlem1

Step	Hyp	Ref	Expression
1		ivth.1	. . . . 5 |- ( ph -> A e. RR )
2	1	rexrd 10089	. . . 4 |- ( ph -> A e. RR* )
3		ivth.2	. . . . 5 |- ( ph -> B e. RR )
4	3	rexrd 10089	. . . 4 |- ( ph -> B e. RR* )
5		ivth.4	. . . . 5 |- ( ph -> A < B )
6	1, 3, 5	ltled 10185	. . . 4 |- ( ph -> A <_ B )
7		lbicc2 12288	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1326	. . 3 |- ( ph -> A e. ( A [,] B ) )
9		ivth.8	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
10	9	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
11		fveq2 6191	. . . . . . 7 |- ( x = A -> ( F ` x ) = ( F ` A ) )
12	11	eleq1d 2686	. . . . . 6 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
13	12	rspcv 3305	. . . . 5 |- ( A e. ( A [,] B ) -> ( A. x e. ( A [,] B ) ( F ` x ) e. RR -> ( F ` A ) e. RR ) )
14	8, 10, 13	sylc 65	. . . 4 |- ( ph -> ( F ` A ) e. RR )
15		ivth.3	. . . 4 |- ( ph -> U e. RR )
16		ivth.9	. . . . 5 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
17	16	simpld 475	. . . 4 |- ( ph -> ( F ` A ) < U )
18	14, 15, 17	ltled 10185	. . 3 |- ( ph -> ( F ` A ) <_ U )
19	11	breq1d 4663	. . . 4 |- ( x = A -> ( ( F ` x ) <_ U <-> ( F ` A ) <_ U ) )
20		ivth.10	. . . 4 |- S = { x e. ( A [,] B ) | ( F ` x ) <_ U }
21	19, 20	elrab2 3366	. . 3 |- ( A e. S <-> ( A e. ( A [,] B ) /\ ( F ` A ) <_ U ) )
22	8, 18, 21	sylanbrc 698	. 2 |- ( ph -> A e. S )
23		ssrab2 3687	. . . . . 6 |- { x e. ( A [,] B ) | ( F ` x ) <_ U } C_ ( A [,] B )
24	20, 23	eqsstri 3635	. . . . 5 |- S C_ ( A [,] B )
25	24	sseli 3599	. . . 4 |- ( z e. S -> z e. ( A [,] B ) )
26		iccleub 12229	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ z e. ( A [,] B ) ) -> z <_ B )
27	26	3expia 1267	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( z e. ( A [,] B ) -> z <_ B ) )
28	2, 4, 27	syl2anc 693	. . . 4 |- ( ph -> ( z e. ( A [,] B ) -> z <_ B ) )
29	25, 28	syl5 34	. . 3 |- ( ph -> ( z e. S -> z <_ B ) )
30	29	ralrimiv 2965	. 2 |- ( ph -> A. z e. S z <_ B )
31	22, 30	jca 554	1 |- ( ph -> ( A e. S /\ A. z e. S z <_ B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178   -cn->ccncf 22679

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-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  ivthlem2  23221  ivthlem3  23222