Theorem ivthinclemlr 13015

Description: Lemma for ivthinc 13021. 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 480	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A e. RR )
3		ivth.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> B e. RR )
5		ivth.3	. . . . . 6 |- ( ph -> U e. RR )
6	5	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> U e. RR )
7		ivth.4	. . . . . 6 |- ( ph -> A < B )
8	7	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ q e. L ) -> A < B )
9		ivth.5	. . . . . 6 |- ( ph -> ( A [,] B ) C_ D )
10	9	ad2antrr 480	. . . . 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 480	. . . . 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 469	. . . . . 6 |- ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 469	. . . . 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 480	. . . . 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 473	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
20	19	adantllr 473	. . . . 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 13014	. . . 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 524	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. ( A [,] B ) )
27		fveq2 5468	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
28	27	eleq1d 2226	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
29	13	ralrimiva 2530	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
30	29	ad3antrrr 484	. . . . . . 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 2821	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) e. RR )
32		fveq2 5468	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
33	32	eleq1d 2226	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
34		fveq2 5468	. . . . . . . . . . 11 |- ( w = r -> ( F ` w ) = ( F ` r ) )
35	34	breq1d 3975	. . . . . . . . . 10 |- ( w = r -> ( ( F ` w ) < U <-> ( F ` r ) < U ) )
36	35, 21	elrab2 2871	. . . . . . . . 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 481	. . . . . . 7 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> r e. ( A [,] B ) )
39	33, 30, 38	rspcdva 2821	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) e. RR )
40	5	ad3antrrr 484	. . . . . 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 3969	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
43		fveq2 5468	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
44	43	breq2d 3977	. . . . . . . . 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 3968	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
47	27	breq1d 3975	. . . . . . . . . . 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 2457	. . . . . . . . 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 2530	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
52	51	ralrimiva 2530	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
53	52	ad3antrrr 484	. . . . . . . . 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 2821	. . . . . . . 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 2821	. . . . . . 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 481	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` r ) < U )
59	31, 39, 40, 56, 58	lttrd 8001	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < U )
60		fveq2 5468	. . . . . . 7 |- ( w = q -> ( F ` w ) = ( F ` q ) )
61	60	breq1d 3975	. . . . . 6 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
62	61, 21	elrab2 2871	. . . . 5 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
63	26, 59, 62	sylanbrc 414	. . . 4 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. L )
64	63	rexlimdva2 2577	. . 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 2530	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 1335   e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439   C_ wss 3102   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731   < clt 7912   [,]cicc 9795   -cn->ccncf 12957

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-map 6595  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-icc 9799  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-cncf 12958

This theorem is referenced by:  ivthinclemex  13020