Theorem ivthinclemum 15626

Description: Lemma for ivthinc 15634. 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 8339	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8339	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8408	. . 3 |- ( ph -> A <_ B )
7		ubicc2 10337	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1274	. 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 5675	. . . . 5 |- ( w = B -> ( F ` w ) = ( F ` B ) )
12	11	breq2d 4126	. . . 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 2979	. . 3 |- ( B e. R <-> ( B e. ( A [,] B ) /\ U < ( F ` B ) ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> B e. R )
16		eleq1 2297	. . 3 |- ( r = B -> ( r e. R <-> B e. R ) )
17	16	rspcev 2923	. 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 1398   e. wcel 2205   E.wrex 2523   {crab 2526   C_ wss 3214   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   RR*cxr 8323   < clt 8324   <_ cle 8325   [,]cicc 10243   -cn->ccncf 15561

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-lttrn 8257

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-icc 10247

This theorem is referenced by:  ivthinclemex  15633