Theorem ivthinclemloc 13269

Description: Lemma for ivthinc 13271. Locatedness. (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
ivthinclemloc	|- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )

Distinct variable groups:   A, r, w   x, A, y, r   B, r, w   x, B, y   w, F   x, F, y   w, U   ph, q, r, x, y   w, q

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

Proof of Theorem ivthinclemloc

Step	Hyp	Ref	Expression
1		simpr 109	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q < r )
2		breq2 3986	. . . . . . . 8 |- ( y = r -> ( q < y <-> q < r ) )
3		fveq2 5486	. . . . . . . . 9 |- ( y = r -> ( F ` y ) = ( F ` r ) )
4	3	breq2d 3994	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
5	2, 4	imbi12d 233	. . . . . . 7 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
6		breq1 3985	. . . . . . . . . 10 |- ( x = q -> ( x < y <-> q < y ) )
7		fveq2 5486	. . . . . . . . . . 11 |- ( x = q -> ( F ` x ) = ( F ` q ) )
8	7	breq1d 3992	. . . . . . . . . 10 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
9	6, 8	imbi12d 233	. . . . . . . . 9 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
10	9	ralbidv 2466	. . . . . . . 8 |- ( x = q -> ( A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) <-> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
11		ivthinc.i	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
12	11	expr 373	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
13	12	ralrimiva 2539	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
14	13	ralrimiva 2539	. . . . . . . . 9 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
15	14	ad2antrr 480	. . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
16		simplrl 525	. . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q e. ( A [,] B ) )
17	10, 15, 16	rspcdva 2835	. . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) )
18		simplrr 526	. . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> r e. ( A [,] B ) )
19	5, 17, 18	rspcdva 2835	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( q < r -> ( F ` q ) < ( F ` r ) ) )
20	1, 19	mpd 13	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) < ( F ` r ) )
21	7	eleq1d 2235	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
22		ivth.8	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
23	22	ralrimiva 2539	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
24	23	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
25	21, 24, 16	rspcdva 2835	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) e. RR )
26		fveq2 5486	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
27	26	eleq1d 2235	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
28	27, 24, 18	rspcdva 2835	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` r ) e. RR )
29		ivth.3	. . . . . . 7 |- ( ph -> U e. RR )
30	29	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> U e. RR )
31		axltwlin 7966	. . . . . 6 |- ( ( ( F ` q ) e. RR /\ ( F ` r ) e. RR /\ U e. RR ) -> ( ( F ` q ) < ( F ` r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) ) )
32	25, 28, 30, 31	syl3anc 1228	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < ( F ` r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) ) )
33	20, 32	mpd 13	. . . 4 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) )
34	16	adantr 274	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. ( A [,] B ) )
35		simpr 109	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> ( F ` q ) < U )
36		fveq2 5486	. . . . . . . . 9 |- ( w = q -> ( F ` w ) = ( F ` q ) )
37	36	breq1d 3992	. . . . . . . 8 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
38		ivthinclem.l	. . . . . . . 8 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
39	37, 38	elrab2 2885	. . . . . . 7 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
40	34, 35, 39	sylanbrc 414	. . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. L )
41	40	ex 114	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < U -> q e. L ) )
42	18	adantr 274	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. ( A [,] B ) )
43		simpr 109	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> U < ( F ` r ) )
44		fveq2 5486	. . . . . . . . 9 |- ( w = r -> ( F ` w ) = ( F ` r ) )
45	44	breq2d 3994	. . . . . . . 8 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
46		ivthinclem.r	. . . . . . . 8 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
47	45, 46	elrab2 2885	. . . . . . 7 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
48	42, 43, 47	sylanbrc 414	. . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. R )
49	48	ex 114	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( U < ( F ` r ) -> r e. R ) )
50	41, 49	orim12d 776	. . . 4 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( ( F ` q ) < U \/ U < ( F ` r ) ) -> ( q e. L \/ r e. R ) ) )
51	33, 50	mpd 13	. . 3 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( q e. L \/ r e. R ) )
52	51	ex 114	. 2 |- ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) -> ( q < r -> ( q e. L \/ r e. R ) ) )
53	52	ralrimivva 2548	1 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1343   e. wcel 2136   A.wral 2444   {crab 2448   C_ wss 3116   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   < clt 7933   [,]cicc 9827   -cn->ccncf 13207

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltwlin 7866

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-iota 5153  df-fv 5196  df-pnf 7935  df-mnf 7936  df-ltxr 7938

This theorem is referenced by:  ivthinclemex  13270