Theorem ivthinclemlm 15516

Description: Lemma for ivthinc 15525. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)

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 ) ) )
ivthinc.i	|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
ivthinclem.l	|- L = { w e. ( A [,] B ) | ( F ` w ) < U }
ivthinclem.r	|- R = { w e. ( A [,] B ) | U < ( F ` w ) }

Assertion

Ref	Expression
ivthinclemlm	|- ( ph -> E. q e. ( A [,] B ) q e. L )

Distinct variable groups:   A, q   w, A   B, q   w, B   w, F   L, q   w, U

Allowed substitution hints:   ph( x, y, w, q)   A( x, y)   B( x, y)   D( x, y, w, q)   R( x, y, w, q)   U( x, y, q)   F( x, y, q)   L( x, y, w)

Proof of Theorem ivthinclemlm

Step	Hyp	Ref	Expression
1		ivth.1	. . . 4 |- ( ph -> A e. RR )
2	1	rexrd 8325	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8325	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8394	. . 3 |- ( ph -> A <_ B )
7		lbicc2 10320	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1274	. 2 |- ( ph -> A e. ( A [,] B ) )
9		ivth.9	. . . 4 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
10	9	simpld 112	. . 3 |- ( ph -> ( F ` A ) < U )
11		fveq2 5672	. . . . 5 |- ( w = A -> ( F ` w ) = ( F ` A ) )
12	11	breq1d 4121	. . . 4 |- ( w = A -> ( ( F ` w ) < U <-> ( F ` A ) < U ) )
13		ivthinclem.l	. . . 4 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
14	12, 13	elrab2 2978	. . 3 |- ( A e. L <-> ( A e. ( A [,] B ) /\ ( F ` A ) < U ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> A e. L )
16		eleq1 2297	. . 3 |- ( q = A -> ( q e. L <-> A e. L ) )
17	16	rspcev 2923	. 2 |- ( ( A e. ( A [,] B ) /\ A e. L ) -> E. q e. ( A [,] B ) q e. L )
18	8, 15, 17	syl2anc 411	1 |- ( ph -> E. q e. ( A [,] B ) q e. L )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   {crab 2526   C_ wss 3213   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   CCcc 8127   RRcr 8128   RR*cxr 8309   < clt 8310   <_ cle 8311   [,]cicc 10227   -cn->ccncf 15452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-pre-ltirr 8241  ax-pre-lttrn 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-icc 10231

This theorem is referenced by:  ivthinclemex  15524