Theorem ivthinclemlr 15080

Description: Lemma for ivthinc 15086. 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 488	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A e. RR )
3		ivth.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> B e. RR )
5		ivth.3	. . . . . 6 |- ( ph -> U e. RR )
6	5	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> U e. RR )
7		ivth.4	. . . . . 6 |- ( ph -> A < B )
8	7	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A < B )
9		ivth.5	. . . . . 6 |- ( ph -> ( A [,] B ) C_ D )
10	9	ad2antrr 488	. . . . 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 488	. . . . 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 477	. . . . . 6 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 477	. . . . 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 488	. . . . 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 481	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
20	19	adantllr 481	. . . . 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 110	. . . . 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 15079	. . . 4 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> E. r e. L q < r )
25	24	ex 115	. . 3 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( q e. L -> E. r e. L q < r ) )
26		simpllr 534	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. ( A [,] B ) )
27		fveq2 5575	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
28	27	eleq1d 2273	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
29	13	ralrimiva 2578	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
30	29	ad3antrrr 492	. . . . . . 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 2881	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) e. RR )
32		fveq2 5575	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
33	32	eleq1d 2273	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
34		fveq2 5575	. . . . . . . . . . 11 |- ( w = r -> ( F ` w ) = ( F ` r ) )
35	34	breq1d 4053	. . . . . . . . . 10 |- ( w = r -> ( ( F ` w ) < U <-> ( F ` r ) < U ) )
36	35, 21	elrab2 2931	. . . . . . . . 9 |- ( r e. L <-> ( r e. ( A [,] B ) /\ ( F ` r ) < U ) )
37	36	simplbi 274	. . . . . . . 8 |- ( r e. L -> r e. ( A [,] B ) )
38	37	ad2antlr 489	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> r e. ( A [,] B ) )
39	33, 30, 38	rspcdva 2881	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) e. RR )
40	5	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> U e. RR )
41		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q < r )
42		breq2 4047	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
43		fveq2 5575	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
44	43	breq2d 4055	. . . . . . . . 9 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
45	42, 44	imbi12d 234	. . . . . . . 8 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
46		breq1 4046	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
47	27	breq1d 4053	. . . . . . . . . . 11 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
48	46, 47	imbi12d 234	. . . . . . . . . 10 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
49	48	ralbidv 2505	. . . . . . . . 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 375	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
51	50	ralrimiva 2578	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
52	51	ralrimiva 2578	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
53	52	ad3antrrr 492	. . . . . . . . 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 2881	. . . . . . . 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 2881	. . . . . . 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 275	. . . . . . 7 |- ( r e. L -> ( F ` r ) < U )
58	57	ad2antlr 489	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) < U )
59	31, 39, 40, 56, 58	lttrd 8197	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < U )
60		fveq2 5575	. . . . . . 7 |- ( w = q -> ( F ` w ) = ( F ` q ) )
61	60	breq1d 4053	. . . . . 6 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
62	61, 21	elrab2 2931	. . . . 5 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
63	26, 59, 62	sylanbrc 417	. . . 4 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. L )
64	63	rexlimdva2 2625	. . 3 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( E. r e. L q < r -> q e. L ) )
65	25, 64	impbid 129	. 2 |- ( ( ph /\ q e. ( A [,] B ) ) -> ( q e. L <-> E. r e. L q < r ) )
66	65	ralrimiva 2578	1 |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   C_ wss 3165   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   CCcc 7922   RRcr 7923   < clt 8106   [,]cicc 10012   -cn->ccncf 15013

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-map 6736  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-icc 10016  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-cncf 15014

This theorem is referenced by:  ivthinclemex  15085