Theorem ivthinclemum 14789

Description: Lemma for ivthinc 14797. 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 8069	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8069	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8138	. . 3 |- ( ph -> A <_ B )
7		ubicc2 10051	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1249	. 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 5554	. . . . 5 |- ( w = B -> ( F ` w ) = ( F ` B ) )
12	11	breq2d 4041	. . . 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 2919	. . 3 |- ( B e. R <-> ( B e. ( A [,] B ) /\ U < ( F ` B ) ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> B e. R )
16		eleq1 2256	. . 3 |- ( r = B -> ( r e. R <-> B e. R ) )
17	16	rspcev 2864	. 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 1364   e. wcel 2164   E.wrex 2473   {crab 2476   C_ wss 3153   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   RR*cxr 8053   < clt 8054   <_ cle 8055   [,]cicc 9957   -cn->ccncf 14725

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984  ax-pre-lttrn 7986

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-icc 9961

This theorem is referenced by:  ivthinclemex  14796