Theorem ivthinclemlm 13406

Description: Lemma for ivthinc 13415. The lower 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
ivthinclemlm	|- ( ph -> E. q e. ( A [,] B ) q e. L )

Distinct variable groups:   A, q   w, A   B, q   w, B   w, F   L, q   w, U

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

Proof of Theorem ivthinclemlm

Step	Hyp	Ref	Expression
1		ivth.1	. . . 4 |- ( ph -> A e. RR )
2	1	rexrd 7969	. . 3 |- ( ph -> A e. RR* )
3		ivth.2	. . . 4 |- ( ph -> B e. RR )
4	3	rexrd 7969	. . 3 |- ( ph -> B e. RR* )
5		ivth.4	. . . 4 |- ( ph -> A < B )
6	1, 3, 5	ltled 8038	. . 3 |- ( ph -> A <_ B )
7		lbicc2 9941	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
8	2, 4, 6, 7	syl3anc 1233	. 2 |- ( ph -> A e. ( A [,] B ) )
9		ivth.9	. . . 4 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
10	9	simpld 111	. . 3 |- ( ph -> ( F ` A ) < U )
11		fveq2 5496	. . . . 5 |- ( w = A -> ( F ` w ) = ( F ` A ) )
12	11	breq1d 3999	. . . 4 |- ( w = A -> ( ( F ` w ) < U <-> ( F ` A ) < U ) )
13		ivthinclem.l	. . . 4 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
14	12, 13	elrab2 2889	. . 3 |- ( A e. L <-> ( A e. ( A [,] B ) /\ ( F ` A ) < U ) )
15	8, 10, 14	sylanbrc 415	. 2 |- ( ph -> A e. L )
16		eleq1 2233	. . 3 |- ( q = A -> ( q e. L <-> A e. L ) )
17	16	rspcev 2834	. 2 |- ( ( A e. ( A [,] B ) /\ A e. L ) -> E. q e. ( A [,] B ) q e. L )
18	8, 15, 17	syl2anc 409	1 |- ( ph -> E. q e. ( A [,] B ) q e. L )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   RR*cxr 7953   < clt 7954   <_ cle 7955   [,]cicc 9848   -cn->ccncf 13351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852

This theorem is referenced by:  ivthinclemex  13414