Theorem ivthinclemlm 14273

Description: Lemma for ivthinc 14282. 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 8010	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8010	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8079	. . 3 |- ( ph -> A <_ B )
7		lbicc2 9987	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1238	. 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 5517	. . . . 5 |- ( w = A -> ( F ` w ) = ( F ` A ) )
12	11	breq1d 4015	. . . 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 2898	. . 3 |- ( A e. L <-> ( A e. ( A [,] B ) /\ ( F ` A ) < U ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> A e. L )
16		eleq1 2240	. . 3 |- ( q = A -> ( q e. L <-> A e. L ) )
17	16	rspcev 2843	. 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 1353   e. wcel 2148   E.wrex 2456   {crab 2459   C_ wss 3131   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   RR*cxr 7994   < clt 7995   <_ cle 7996   [,]cicc 9894   -cn->ccncf 14218

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-pre-ltirr 7926  ax-pre-lttrn 7928

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-icc 9898

This theorem is referenced by:  ivthinclemex  14281