Theorem ivthinclemum 13984

Description: Lemma for ivthinc 13992. The upper 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
ivthinclemum	|- ( ph -> E. r e. ( A [,] B ) r e. R )

Distinct variable groups:   A, r   w, A   B, r   w, B   w, F   R, r   w, U

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

Proof of Theorem ivthinclemum

Step	Hyp	Ref	Expression
1		ivth.1	. . . 4 |- ( ph -> A e. RR )
2	1	rexrd 8003	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8003	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8072	. . 3 |- ( ph -> A <_ B )
7		ubicc2 9981	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1238	. 2 |- ( ph -> B e. ( A [,] B ) )
9		ivth.9	. . . 4 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
10	9	simprd 114	. . 3 |- ( ph -> U < ( F ` B ) )
11		fveq2 5514	. . . . 5 |- ( w = B -> ( F ` w ) = ( F ` B ) )
12	11	breq2d 4014	. . . 4 |- ( w = B -> ( U < ( F ` w ) <-> U < ( F ` B ) ) )
13		ivthinclem.r	. . . 4 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
14	12, 13	elrab2 2896	. . 3 |- ( B e. R <-> ( B e. ( A [,] B ) /\ U < ( F ` B ) ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> B e. R )
16		eleq1 2240	. . 3 |- ( r = B -> ( r e. R <-> B e. R ) )
17	16	rspcev 2841	. 2 |- ( ( B e. ( A [,] B ) /\ B e. R ) -> E. r e. ( A [,] B ) r e. R )
18	8, 15, 17	syl2anc 411	1 |- ( ph -> E. r e. ( A [,] B ) r e. R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   C_ wss 3129   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   CCcc 7806   RRcr 7807   RR*cxr 7987   < clt 7988   <_ cle 7989   [,]cicc 9887   -cn->ccncf 13928

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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-pre-ltirr 7920  ax-pre-lttrn 7922

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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-icc 9891

This theorem is referenced by:  ivthinclemex  13991