Theorem ivthinclemlr 15194

Description: Lemma for ivthinc 15200. 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 15193	. . . 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 534	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> q e. ( A [,] B ) )
27		fveq2 5594	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
28	27	eleq1d 2275	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
29	13	ralrimiva 2580	. . . . . . . 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 2886	. . . . . 6 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) e. RR )
32		fveq2 5594	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
33	32	eleq1d 2275	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
34		fveq2 5594	. . . . . . . . . . 11 |- ( w = r -> ( F ` w ) = ( F ` r ) )
35	34	breq1d 4064	. . . . . . . . . 10 |- ( w = r -> ( ( F ` w ) < U <-> ( F ` r ) < U ) )
36	35, 21	elrab2 2936	. . . . . . . . 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 2886	. . . . . 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 4058	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
43		fveq2 5594	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
44	43	breq2d 4066	. . . . . . . . 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 4057	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
47	27	breq1d 4064	. . . . . . . . . . 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 2507	. . . . . . . . 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 2580	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
52	51	ralrimiva 2580	. . . . . . . . . 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 2886	. . . . . . . 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 2886	. . . . . . 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 8228	. . . . 5 |- ( ( ( ( ph /\ q e. ( A [,] B ) ) /\ r e. L ) /\ q < r ) -> ( F ` q ) < U )
60		fveq2 5594	. . . . . . 7 |- ( w = q -> ( F ` w ) = ( F ` q ) )
61	60	breq1d 4064	. . . . . 6 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
62	61, 21	elrab2 2936	. . . . 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 2627	. . 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 2580	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 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   C_ wss 3170   class class class wbr 4054   ` cfv 5285  (class class class)co 5962   CCcc 7953   RRcr 7954   < clt 8137   [,]cicc 10043   -cn->ccncf 15127

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-map 6755  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-rp 9806  df-icc 10047  df-seqfrec 10625  df-exp 10716  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395  df-cncf 15128

This theorem is referenced by:  ivthinclemex  15199