Theorem ivthinclemlr 13409

Description: Lemma for ivthinc 13415. The lower cut is rounded. (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
ivthinclemlr	|- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )

Distinct variable groups:   A, r, w   x, A, y, r   B, r, w   x, B, y   w, F   x, F, y   L, r, x, 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( w, q)

Proof of Theorem ivthinclemlr

Step	Hyp	Ref	Expression
1		ivth.1	. . . . . 6 |- ( ph -> A e. RR )
2	1	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A e. RR )
3		ivth.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> B e. RR )
5		ivth.3	. . . . . 6 |- ( ph -> U e. RR )
6	5	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> U e. RR )
7		ivth.4	. . . . . 6 |- ( ph -> A < B )
8	7	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A < B )
9		ivth.5	. . . . . 6 |- ( ph -> ( A [,] B ) C_ D )
10	9	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> ( A [,] B ) C_ D )
11		ivth.7	. . . . . 6 |- ( ph -> F e. ( D -cn-> CC ) )
12	11	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> F e. ( D -cn-> CC ) )
13		ivth.8	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
14	13	adantlr 474	. . . . . 6 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 474	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
16		ivth.9	. . . . . 6 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
17	16	ad2antrr 485	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
18		ivthinc.i	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
19	18	adantllr 478	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
20	19	adantllr 478	. . . . 5 |- ( ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
21		ivthinclem.l	. . . . 5 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
22		ivthinclem.r	. . . . 5 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
23		simpr 109	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> q e. L )
24	2, 4, 6, 8, 10, 12, 15, 17, 20, 21, 22, 23	ivthinclemlopn 13408	. . . 4 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> E. r e. L q < r )
25	24	ex 114	. . 3 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( q e. L -> E. r e. L q < r ) )
26		simpllr 529	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. ( A [,] B ) )
27		fveq2 5496	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
28	27	eleq1d 2239	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
29	13	ralrimiva 2543	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
30	29	ad3antrrr 489	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
31	28, 30, 26	rspcdva 2839	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) e. RR )
32		fveq2 5496	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
33	32	eleq1d 2239	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
34		fveq2 5496	. . . . . . . . . . 11 |- ( w = r -> ( F ` w ) = ( F ` r ) )
35	34	breq1d 3999	. . . . . . . . . 10 |- ( w = r -> ( ( F ` w ) < U <-> ( F ` r ) < U ) )
36	35, 21	elrab2 2889	. . . . . . . . 9 |- ( r e. L <-> ( r e. ( A [,] B ) /\ ( F ` r ) < U ) )
37	36	simplbi 272	. . . . . . . 8 |- ( r e. L -> r e. ( A [,] B ) )
38	37	ad2antlr 486	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> r e. ( A [,] B ) )
39	33, 30, 38	rspcdva 2839	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) e. RR )
40	5	ad3antrrr 489	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> U e. RR )
41		simpr 109	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q < r )
42		breq2 3993	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
43		fveq2 5496	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
44	43	breq2d 4001	. . . . . . . . 9 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
45	42, 44	imbi12d 233	. . . . . . . 8 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
46		breq1 3992	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
47	27	breq1d 3999	. . . . . . . . . . 11 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
48	46, 47	imbi12d 233	. . . . . . . . . 10 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
49	48	ralbidv 2470	. . . . . . . . 9 |- ( x = q -> ( A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) <-> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
50	18	expr 373	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
51	50	ralrimiva 2543	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
52	51	ralrimiva 2543	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
53	52	ad3antrrr 489	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	49, 53, 26	rspcdva 2839	. . . . . . . 8 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) )
55	45, 54, 38	rspcdva 2839	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( q < r -> ( F ` q ) < ( F ` r ) ) )
56	41, 55	mpd 13	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < ( F ` r ) )
57	36	simprbi 273	. . . . . . 7 |- ( r e. L -> ( F ` r ) < U )
58	57	ad2antlr 486	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) < U )
59	31, 39, 40, 56, 58	lttrd 8045	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < U )
60		fveq2 5496	. . . . . . 7 |- ( w = q -> ( F ` w ) = ( F ` q ) )
61	60	breq1d 3999	. . . . . 6 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
62	61, 21	elrab2 2889	. . . . 5 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
63	26, 59, 62	sylanbrc 415	. . . 4 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. L )
64	63	rexlimdva2 2590	. . 3 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( E. r e. L q < r -> q e. L ) )
65	25, 64	impbid 128	. 2 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( q e. L <-> E. r e. L q < r ) )
66	65	ralrimiva 2543	1 |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  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-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-icc 9852  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-cncf 13352

This theorem is referenced by:  ivthinclemex  13414