Theorem ivthinclemur 15186

Description: Lemma for ivthinc 15190. 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 15185	. . . 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 5589	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
29	28	eleq1d 2275	. . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
30	13	ralrimiva 2580	. . . . . . . 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 5589	. . . . . . . . . . 11 |- ( w = q -> ( F ` w ) = ( F ` q ) )
33	32	breq2d 4063	. . . . . . . . . 10 |- ( w = q -> ( U < ( F ` w ) <-> U < ( F ` q ) ) )
34	33, 22	elrab2 2936	. . . . . . . . 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 2886	. . . . . 6 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> ( F ` q ) e. RR )
38		fveq2 5589	. . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
39	38	eleq1d 2275	. . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
40	39, 31, 26	rspcdva 2886	. . . . . 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 4055	. . . . . . . . 9 |- ( y = r -> ( q < y <-> q < r ) )
45		fveq2 5589	. . . . . . . . . 10 |- ( y = r -> ( F ` y ) = ( F ` r ) )
46	45	breq2d 4063	. . . . . . . . 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 4054	. . . . . . . . . . 11 |- ( x = q -> ( x < y <-> q < y ) )
49	28	breq1d 4061	. . . . . . . . . . 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 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 ) ) ) )
52	18	expr 375	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
53	52	ralrimiva 2580	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
54	53	ralrimiva 2580	. . . . . . . . . 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 2886	. . . . . . . 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 2886	. . . . . . 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 8218	. . . . 5 |- ( ( ( ( ph /\ r e. ( A [,] B ) ) /\ q e. R ) /\ q < r ) -> U < ( F ` r ) )
60		fveq2 5589	. . . . . . 7 |- ( w = r -> ( F ` w ) = ( F ` r ) )
61	60	breq2d 4063	. . . . . 6 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
62	61, 22	elrab2 2936	. . . . 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 2627	. . 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 2580	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 2177   A.wral 2485   E.wrex 2486   {crab 2489   C_ wss 3170   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   CCcc 7943   RRcr 7944   < clt 8127   [,]cicc 10033   -cn->ccncf 15117

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 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

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 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-map 6750  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-rp 9796  df-icc 10037  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-cncf 15118

This theorem is referenced by:  ivthinclemex  15189