Theorem ivthinclemlr 15502

Description: Lemma for ivthinc 15508. 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 15501	. . . 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 536	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. ( A [,] B ) )
27		fveq2 5670	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
28	27	eleq1d 2301	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
29	13	ralrimiva 2615	. . . . . . . 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 2926	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) e. RR )
32		fveq2 5670	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
33	32	eleq1d 2301	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
34		fveq2 5670	. . . . . . . . . . 11 |- ( w = r -> ( F ` w ) = ( F ` r ) )
35	34	breq1d 4119	. . . . . . . . . 10 |- ( w = r -> ( ( F ` w ) < U <-> ( F ` r ) < U ) )
36	35, 21	elrab2 2976	. . . . . . . . 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 2926	. . . . . 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 4113	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
43		fveq2 5670	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
44	43	breq2d 4121	. . . . . . . . 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 4112	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
47	27	breq1d 4119	. . . . . . . . . . 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 2542	. . . . . . . . 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 2615	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
52	51	ralrimiva 2615	. . . . . . . . . 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 2926	. . . . . . . 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 2926	. . . . . . 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 8399	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < U )
60		fveq2 5670	. . . . . . 7 |- ( w = q -> ( F ` w ) = ( F ` q ) )
61	60	breq1d 4119	. . . . . 6 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
62	61, 21	elrab2 2976	. . . . 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 2663	. . 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 2615	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 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   < clt 8308   [,]cicc 10224   -cn->ccncf 15435

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-icc 10228  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-cncf 15436

This theorem is referenced by:  ivthinclemex  15507