Theorem ivthinclemloc 13413

Description: Lemma for ivthinc 13415. 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 3993	. . . . . . . 8 |- ( y = r -> ( q < y <-> q < r ) )
3		fveq2 5496	. . . . . . . . 9 |- ( y = r -> ( F ` y ) = ( F ` r ) )
4	3	breq2d 4001	. . . . . . . 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 3992	. . . . . . . . . 10 |- ( x = q -> ( x < y <-> q < y ) )
7		fveq2 5496	. . . . . . . . . . 11 |- ( x = q -> ( F ` x ) = ( F ` q ) )
8	7	breq1d 3999	. . . . . . . . . 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 2470	. . . . . . . 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 2543	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
14	13	ralrimiva 2543	. . . . . . . . 9 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
15	14	ad2antrr 485	. . . . . . . 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 530	. . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q e. ( A [,] B ) )
17	10, 15, 16	rspcdva 2839	. . . . . . 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 531	. . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> r e. ( A [,] B ) )
19	5, 17, 18	rspcdva 2839	. . . . . 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 2239	. . . . . . 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 2543	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
24	23	ad2antrr 485	. . . . . . 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 2839	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) e. RR )
26		fveq2 5496	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
27	26	eleq1d 2239	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
28	27, 24, 18	rspcdva 2839	. . . . . 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 485	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> U e. RR )
31		axltwlin 7987	. . . . . 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 1233	. . . . 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 5496	. . . . . . . . 9 |- ( w = q -> ( F ` w ) = ( F ` q ) )
37	36	breq1d 3999	. . . . . . . 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 2889	. . . . . . 7 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
40	34, 35, 39	sylanbrc 415	. . . . . 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 5496	. . . . . . . . 9 |- ( w = r -> ( F ` w ) = ( F ` r ) )
45	44	breq2d 4001	. . . . . . . 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 2889	. . . . . . 7 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
48	42, 43, 47	sylanbrc 415	. . . . . 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 781	. . . 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 2552	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 703   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   < clt 7954   [,]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-ltwlin 7887

This theorem depends on definitions:  df-bi 116  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-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-xp 4617  df-iota 5160  df-fv 5206  df-pnf 7956  df-mnf 7957  df-ltxr 7959

This theorem is referenced by:  ivthinclemex  13414