Theorem ivthinclemur 15111

Description: Lemma for ivthinc 15115. The upper 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
ivthinclemur	|- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )

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

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

Proof of Theorem ivthinclemur

Step	Hyp	Ref	Expression
1		ivth.1	. . . . . 6 |- ( ph -> A e. RR )
2	1	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> A e. RR )
3		ivth.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> B e. RR )
5		ivth.3	. . . . . 6 |- ( ph -> U e. RR )
6	5	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> U e. RR )
7		ivth.4	. . . . . 6 |- ( ph -> A < B )
8	7	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> A < B )
9		ivth.5	. . . . . 6 |- ( ph -> ( A [,] B ) C_ D )
10	9	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> ( A [,] B ) C_ D )
11		ivth.7	. . . . . 6 |- ( ph -> F e. ( D -cn-> CC ) )
12	11	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> F e. ( D -cn-> CC ) )
13		ivth.8	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
14	13	adantlr 477	. . . . . 6 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 477	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) /\ 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 /\ r e. ( A [,] B ) ) /\ r e. R ) -> ( ( 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 /\ r e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
20	19	adantllr 481	. . . . 5 |- ( ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) /\ 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 /\ r e. ( A [,] B ) ) /\ r e. R ) -> r e. R )
24	2, 4, 6, 8, 10, 12, 15, 17, 20, 21, 22, 23	ivthinclemuopn 15110	. . . 4 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> E. q e. R q < r )
25	24	ex 115	. . 3 |- ( ( ph /\ r e. ( A [,] B ) ) -> ( r e. R -> E. q e. R q < r ) )
26		simpllr 534	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> r e. ( A [,] B ) )
27	5	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U e. RR )
28		fveq2 5576	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
29	28	eleq1d 2274	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
30	13	ralrimiva 2579	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
31	30	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
32		fveq2 5576	. . . . . . . . . . 11 |- ( w = q -> ( F ` w ) = ( F ` q ) )
33	32	breq2d 4056	. . . . . . . . . 10 |- ( w = q -> ( U < ( F ` w ) <-> U < ( F ` q ) ) )
34	33, 22	elrab2 2932	. . . . . . . . 9 |- ( q e. R <-> ( q e. ( A [,] B ) /\ U < ( F ` q ) ) )
35	34	simplbi 274	. . . . . . . 8 |- ( q e. R -> q e. ( A [,] B ) )
36	35	ad2antlr 489	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> q e. ( A [,] B ) )
37	29, 31, 36	rspcdva 2882	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) e. RR )
38		fveq2 5576	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
39	38	eleq1d 2274	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
40	39, 31, 26	rspcdva 2882	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` r ) e. RR )
41	34	simprbi 275	. . . . . . 7 |- ( q e. R -> U < ( F ` q ) )
42	41	ad2antlr 489	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` q ) )
43		simpr 110	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> q < r )
44		breq2 4048	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
45		fveq2 5576	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
46	45	breq2d 4056	. . . . . . . . 9 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
47	44, 46	imbi12d 234	. . . . . . . 8 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
48		breq1 4047	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
49	28	breq1d 4054	. . . . . . . . . . 11 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
50	48, 49	imbi12d 234	. . . . . . . . . 10 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
51	50	ralbidv 2506	. . . . . . . . 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 ) ) ) )
52	18	expr 375	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
53	52	ralrimiva 2579	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	53	ralrimiva 2579	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
55	54	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
56	51, 55, 36	rspcdva 2882	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) )
57	47, 56, 26	rspcdva 2882	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( q < r -> ( F ` q ) < ( F ` r ) ) )
58	43, 57	mpd 13	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) < ( F ` r ) )
59	27, 37, 40, 42, 58	lttrd 8198	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` r ) )
60		fveq2 5576	. . . . . . 7 |- ( w = r -> ( F ` w ) = ( F ` r ) )
61	60	breq2d 4056	. . . . . 6 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
62	61, 22	elrab2 2932	. . . . 5 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
63	26, 59, 62	sylanbrc 417	. . . 4 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> r e. R )
64	63	rexlimdva2 2626	. . 3 |- ( ( ph /\ r e. ( A [,] B ) ) -> ( E. q e. R q < r -> r e. R ) )
65	25, 64	impbid 129	. 2 |- ( ( ph /\ r e. ( A [,] B ) ) -> ( r e. R <-> E. q e. R q < r ) )
66	65	ralrimiva 2579	1 |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   C_ wss 3166   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   < clt 8107   [,]cicc 10013   -cn->ccncf 15042

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-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  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-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-map 6737  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-icc 10017  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-cncf 15043

This theorem is referenced by:  ivthinclemex  15114