Theorem ivthinclemum 15358

Description: Lemma for ivthinc 15366. 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 8228	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8228	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8297	. . 3 |- ( ph -> A <_ B )
7		ubicc2 10219	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1273	. 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 5639	. . . . 5 |- ( w = B -> ( F ` w ) = ( F ` B ) )
12	11	breq2d 4100	. . . 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 2965	. . 3 |- ( B e. R <-> ( B e. ( A [,] B ) /\ U < ( F ` B ) ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> B e. R )
16		eleq1 2294	. . 3 |- ( r = B -> ( r e. R <-> B e. R ) )
17	16	rspcev 2910	. 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 1397   e. wcel 2202   E.wrex 2511   {crab 2514   C_ wss 3200   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   RR*cxr 8212   < clt 8213   <_ cle 8214   [,]cicc 10125   -cn->ccncf 15293

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-icc 10129

This theorem is referenced by:  ivthinclemex  15365