Theorem ivthinclemum 15222

Description: Lemma for ivthinc 15230. 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 8157	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 8157	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8226	. . 3 |- ( ph -> A <_ B )
7		ubicc2 10142	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1250	. 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 5599	. . . . 5 |- ( w = B -> ( F ` w ) = ( F ` B ) )
12	11	breq2d 4071	. . . 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 2939	. . 3 |- ( B e. R <-> ( B e. ( A [,] B ) /\ U < ( F ` B ) ) )
15	8, 10, 14	sylanbrc 417	. 2 |- ( ph -> B e. R )
16		eleq1 2270	. . 3 |- ( r = B -> ( r e. R <-> B e. R ) )
17	16	rspcev 2884	. 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 1373   e. wcel 2178   E.wrex 2487   {crab 2490   C_ wss 3174   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   RR*cxr 8141   < clt 8142   <_ cle 8143   [,]cicc 10048   -cn->ccncf 15157

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072  ax-pre-lttrn 8074

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-icc 10052

This theorem is referenced by:  ivthinclemex  15229