Theorem ivthinclemur 15740

Description: Lemma for ivthinc 15744. 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 492	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> A e. RR )
3		ivth.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> B e. RR )
5		ivth.3	. . . . . 6 |- ( ph -> U e. RR )
6	5	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> U e. RR )
7		ivth.4	. . . . . 6 |- ( ph -> A < B )
8	7	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ r e. R ) -> A < B )
9		ivth.5	. . . . . 6 |- ( ph -> ( A [,] B ) C_ D )
10	9	ad2antrr 492	. . . . 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 492	. . . . 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 481	. . . . . 6 |- ( ( ( ph /\ r e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 481	. . . . 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 492	. . . . 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 485	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
20	19	adantllr 485	. . . . 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 15739	. . . 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 540	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> r e. ( A [,] B ) )
27	5	ad3antrrr 496	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U e. RR )
28		fveq2 5695	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
29	28	eleq1d 2307	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
30	13	ralrimiva 2623	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
31	30	ad3antrrr 496	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
32		fveq2 5695	. . . . . . . . . . 11 |- ( w = q -> ( F ` w ) = ( F ` q ) )
33	32	breq2d 4142	. . . . . . . . . 10 |- ( w = q -> ( U < ( F ` w ) <-> U < ( F ` q ) ) )
34	33, 22	elrab2 2985	. . . . . . . . 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 493	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> q e. ( A [,] B ) )
37	29, 31, 36	rspcdva 2934	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) e. RR )
38		fveq2 5695	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
39	38	eleq1d 2307	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
40	39, 31, 26	rspcdva 2934	. . . . . 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 493	. . . . . 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 4134	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
45		fveq2 5695	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
46	45	breq2d 4142	. . . . . . . . 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 4133	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
49	28	breq1d 4140	. . . . . . . . . . 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 2550	. . . . . . . . 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 2623	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	53	ralrimiva 2623	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
55	54	ad3antrrr 496	. . . . . . . . 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 2934	. . . . . . . 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 2934	. . . . . . 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 8452	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` r ) )
60		fveq2 5695	. . . . . . 7 |- ( w = r -> ( F ` w ) = ( F ` r ) )
61	60	breq2d 4142	. . . . . 6 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
62	61, 22	elrab2 2985	. . . . 5 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
63	26, 59, 62	sylanbrc 421	. . . 4 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> r e. R )
64	63	rexlimdva2 2671	. . 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 2623	1 |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   < clt 8360   [,]cicc 10293   -cn->ccncf 15671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-icc 10297  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-cncf 15672

This theorem is used by:  ivthinclemex  15743