Theorem ivthinclemlm 15308

Description: Lemma for ivthinc 15317. 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 8196	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8196	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8265	. . 3 |- ( ph -> A <_ B )
7		lbicc2 10180	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1271	. 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 5627	. . . . 5 |- ( w = A -> ( F ` w ) = ( F ` A ) )
12	11	breq1d 4093	. . . 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 2962	. . 3 |- ( A e. L <-> ( A e. ( A [,] B ) /\ ( F ` A ) < U ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> A e. L )
16		eleq1 2292	. . 3 |- ( q = A -> ( q e. L <-> A e. L ) )
17	16	rspcev 2907	. 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 1395   e. wcel 2200   E.wrex 2509   {crab 2512   C_ wss 3197   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   RR*cxr 8180   < clt 8181   <_ cle 8182   [,]cicc 10087   -cn->ccncf 15244

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111  ax-pre-lttrn 8113

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-icc 10091

This theorem is referenced by:  ivthinclemex  15316