Theorem ivthinclemur 15226

Description: Lemma for ivthinc 15230. 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 15225	. . . 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 5599	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
29	28	eleq1d 2276	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
30	13	ralrimiva 2581	. . . . . . . 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 5599	. . . . . . . . . . 11 |- ( w = q -> ( F ` w ) = ( F ` q ) )
33	32	breq2d 4071	. . . . . . . . . 10 |- ( w = q -> ( U < ( F ` w ) <-> U < ( F ` q ) ) )
34	33, 22	elrab2 2939	. . . . . . . . 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 2889	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) e. RR )
38		fveq2 5599	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
39	38	eleq1d 2276	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
40	39, 31, 26	rspcdva 2889	. . . . . 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 4063	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
45		fveq2 5599	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
46	45	breq2d 4071	. . . . . . . . 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 4062	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
49	28	breq1d 4069	. . . . . . . . . . 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 2508	. . . . . . . . 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 2581	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	53	ralrimiva 2581	. . . . . . . . . 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 2889	. . . . . . . 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 2889	. . . . . . 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 8233	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` r ) )
60		fveq2 5599	. . . . . . 7 |- ( w = r -> ( F ` w ) = ( F ` r ) )
61	60	breq2d 4071	. . . . . 6 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
62	61, 22	elrab2 2939	. . . . 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 2628	. . 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 2581	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 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   {crab 2490   C_ wss 3174   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   < clt 8142   [,]cicc 10048   -cn->ccncf 15157

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 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-icc 10052  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-cncf 15158

This theorem is referenced by:  ivthinclemex  15229