Theorem ivthinclemur 14818

Description: Lemma for ivthinc 14822. 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 14817	. . . 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 5555	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
29	28	eleq1d 2262	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
30	13	ralrimiva 2567	. . . . . . . 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 5555	. . . . . . . . . . 11 |- ( w = q -> ( F ` w ) = ( F ` q ) )
33	32	breq2d 4042	. . . . . . . . . 10 |- ( w = q -> ( U < ( F ` w ) <-> U < ( F ` q ) ) )
34	33, 22	elrab2 2920	. . . . . . . . 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 2870	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) e. RR )
38		fveq2 5555	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
39	38	eleq1d 2262	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
40	39, 31, 26	rspcdva 2870	. . . . . 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 4034	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
45		fveq2 5555	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
46	45	breq2d 4042	. . . . . . . . 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 4033	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
49	28	breq1d 4040	. . . . . . . . . . 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 2494	. . . . . . . . 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 2567	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	53	ralrimiva 2567	. . . . . . . . . 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 2870	. . . . . . . 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 2870	. . . . . . 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 8147	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` r ) )
60		fveq2 5555	. . . . . . 7 |- ( w = r -> ( F ` w ) = ( F ` r ) )
61	60	breq2d 4042	. . . . . 6 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
62	61, 22	elrab2 2920	. . . . 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 2614	. . 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 2567	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 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3154   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   < clt 8056   [,]cicc 9960   -cn->ccncf 14749

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-rp 9723  df-icc 9964  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-cncf 14750

This theorem is referenced by:  ivthinclemex  14821