Theorem ivthinclemloc 15146

Description: Lemma for ivthinc 15148. 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 110	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q < r )
2		breq2 4049	. . . . . . . 8 |- ( y = r -> ( q < y <-> q < r ) )
3		fveq2 5578	. . . . . . . . 9 |- ( y = r -> ( F ` y ) = ( F ` r ) )
4	3	breq2d 4057	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
5	2, 4	imbi12d 234	. . . . . . 7 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
6		breq1 4048	. . . . . . . . . 10 |- ( x = q -> ( x < y <-> q < y ) )
7		fveq2 5578	. . . . . . . . . . 11 |- ( x = q -> ( F ` x ) = ( F ` q ) )
8	7	breq1d 4055	. . . . . . . . . 10 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
9	6, 8	imbi12d 234	. . . . . . . . 9 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
10	9	ralbidv 2506	. . . . . . . 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 375	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
13	12	ralrimiva 2579	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
14	13	ralrimiva 2579	. . . . . . . . 9 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
15	14	ad2antrr 488	. . . . . . . 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 535	. . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q e. ( A [,] B ) )
17	10, 15, 16	rspcdva 2882	. . . . . . 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 536	. . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> r e. ( A [,] B ) )
19	5, 17, 18	rspcdva 2882	. . . . . 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 2274	. . . . . . 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 2579	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
24	23	ad2antrr 488	. . . . . . 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 2882	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) e. RR )
26		fveq2 5578	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
27	26	eleq1d 2274	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
28	27, 24, 18	rspcdva 2882	. . . . . 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 488	. . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> U e. RR )
31		axltwlin 8142	. . . . . 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 1250	. . . . 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 276	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. ( A [,] B ) )
35		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> ( F ` q ) < U )
36		fveq2 5578	. . . . . . . . 9 |- ( w = q -> ( F ` w ) = ( F ` q ) )
37	36	breq1d 4055	. . . . . . . 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 2932	. . . . . . 7 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
40	34, 35, 39	sylanbrc 417	. . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. L )
41	40	ex 115	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < U -> q e. L ) )
42	18	adantr 276	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. ( A [,] B ) )
43		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> U < ( F ` r ) )
44		fveq2 5578	. . . . . . . . 9 |- ( w = r -> ( F ` w ) = ( F ` r ) )
45	44	breq2d 4057	. . . . . . . 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 2932	. . . . . . 7 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
48	42, 43, 47	sylanbrc 417	. . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. R )
49	48	ex 115	. . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( U < ( F ` r ) -> r e. R ) )
50	41, 49	orim12d 788	. . . 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 115	. 2 |- ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) -> ( q < r -> ( q e. L \/ r e. R ) ) )
53	52	ralrimivva 2588	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 104   \/ wo 710   = wceq 1373   e. wcel 2176   A.wral 2484   {crab 2488   C_ wss 3166   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   < clt 8109   [,]cicc 10015   -cn->ccncf 15075

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-pre-ltwlin 8040

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-xp 4682  df-iota 5233  df-fv 5280  df-pnf 8111  df-mnf 8112  df-ltxr 8114

This theorem is referenced by:  ivthinclemex  15147