Theorem ivthinclemlm 12781

Description: Lemma for ivthinc 12790. 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 7815	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 7815	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 7881	. . 3 |- ( ph -> A <_ B )
7		lbicc2 9767	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1216	. 2 |- ( ph -> A e. ( A [,] B ) )
9		ivth.9	. . . 4 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
10	9	simpld 111	. . 3 |- ( ph -> ( F ` A ) < U )
11		fveq2 5421	. . . . 5 |- ( w = A -> ( F ` w ) = ( F ` A ) )
12	11	breq1d 3939	. . . 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 2843	. . 3 |- ( A e. L <-> ( A e. ( A [,] B ) /\ ( F ` A ) < U ) )
15	8, 10, 14	sylanbrc 413	. 2 |- ( ph -> A e. L )
16		eleq1 2202	. . 3 |- ( q = A -> ( q e. L <-> A e. L ) )
17	16	rspcev 2789	. 2 |- ( ( A e. ( A [,] B ) /\ A e. L ) -> E. q e. ( A [,] B ) q e. L )
18	8, 15, 17	syl2anc 408	1 |- ( ph -> E. q e. ( A [,] B ) q e. L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   E.wrex 2417   {crab 2420   C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   RR*cxr 7799   < clt 7800   <_ cle 7801   [,]cicc 9674   -cn->ccncf 12726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-lttrn 7734

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678

This theorem is referenced by:  ivthinclemex  12789