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Theorem infleinflem1 39586
Description: Lemma for infleinf 39588, case B =/= (/) /\ -oo < inf ( B , RR* , < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
infleinflem1.a |- ( ph -> A C_ RR* )
infleinflem1.b |- ( ph -> B C_ RR* )
infleinflem1.w |- ( ph -> W e. RR+ )
infleinflem1.x |- ( ph -> X e. B )
infleinflem1.i |- ( ph -> X <_ (inf ( B , RR* , < ) +e ( W / 2 ) ) )
infleinflem1.z |- ( ph -> Z e. A )
infleinflem1.l |- ( ph -> Z <_ ( X +e ( W / 2 ) ) )
Assertion
Ref Expression
infleinflem1 |- ( ph -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e W ) )
Proof of Theorem infleinflem1
StepHypRef Expression
1 infleinflem1.a . . . 4 |- ( ph -> A C_ RR* )
2 infxrcl 12163 . . . 4 |- ( A C_ RR* -> inf ( A , RR* , < ) e. RR* )
31, 2syl 17 . . 3 |- ( ph -> inf ( A , RR* , < ) e. RR* )
4 id 22 . . 3 |- (inf ( A , RR* , < ) e. RR* -> inf ( A , RR* , < ) e. RR* )
53, 4syl 17 . 2 |- ( ph -> inf ( A , RR* , < ) e. RR* )
6 infleinflem1.z . . 3 |- ( ph -> Z e. A )
71, 6sseldd 3604 . 2 |- ( ph -> Z e. RR* )
8 infleinflem1.b . . . 4 |- ( ph -> B C_ RR* )
9 infxrcl 12163 . . . 4 |- ( B C_ RR* -> inf ( B , RR* , < ) e. RR* )
108, 9syl 17 . . 3 |- ( ph -> inf ( B , RR* , < ) e. RR* )
11 infleinflem1.w . . . 4 |- ( ph -> W e. RR+ )
12 rpxr 11840 . . . 4 |- ( W e. RR+ -> W e. RR* )
1311, 12syl 17 . . 3 |- ( ph -> W e. RR* )
1410, 13xaddcld 12131 . 2 |- ( ph -> (inf ( B , RR* , < ) +e W ) e. RR* )
15 infxrlb 12164 . . 3 |- ( ( A C_ RR* /\ Z e. A ) -> inf ( A , RR* , < ) <_ Z )
161, 6, 15syl2anc 693 . 2 |- ( ph -> inf ( A , RR* , < ) <_ Z )
17 infleinflem1.x . . . . 5 |- ( ph -> X e. B )
188sselda 3603 . . . . 5 |- ( ( ph /\ X e. B ) -> X e. RR* )
1917, 18mpdan 702 . . . 4 |- ( ph -> X e. RR* )
2011rpred 11872 . . . . . 6 |- ( ph -> W e. RR )
2120rehalfcld 11279 . . . . 5 |- ( ph -> ( W / 2 ) e. RR )
2221rexrd 10089 . . . 4 |- ( ph -> ( W / 2 ) e. RR* )
2319, 22xaddcld 12131 . . 3 |- ( ph -> ( X +e ( W / 2 ) ) e. RR* )
24 infleinflem1.l . . 3 |- ( ph -> Z <_ ( X +e ( W / 2 ) ) )
25 pnfge 11964 . . . . . . 7 |- ( ( X +e ( W / 2 ) ) e. RR* -> ( X +e ( W / 2 ) ) <_ +oo )
2623, 25syl 17 . . . . . 6 |- ( ph -> ( X +e ( W / 2 ) ) <_ +oo )
2726adantr 481 . . . . 5 |- ( ( ph /\ inf ( B , RR* , < ) = +oo ) -> ( X +e ( W / 2 ) ) <_ +oo )
28 oveq1 6657 . . . . . . 7 |- (inf ( B , RR* , < ) = +oo -> (inf ( B , RR* , < ) +e W ) = ( +oo +e W ) )
2928adantl 482 . . . . . 6 |- ( ( ph /\ inf ( B , RR* , < ) = +oo ) -> (inf ( B , RR* , < ) +e W ) = ( +oo +e W ) )
30 rpre 11839 . . . . . . . . . 10 |- ( W e. RR+ -> W e. RR )
31 renemnf 10088 . . . . . . . . . 10 |- ( W e. RR -> W =/= -oo )
3230, 31syl 17 . . . . . . . . 9 |- ( W e. RR+ -> W =/= -oo )
33 xaddpnf2 12058 . . . . . . . . 9 |- ( ( W e. RR* /\ W =/= -oo ) -> ( +oo +e W ) = +oo )
3412, 32, 33syl2anc 693 . . . . . . . 8 |- ( W e. RR+ -> ( +oo +e W ) = +oo )
3511, 34syl 17 . . . . . . 7 |- ( ph -> ( +oo +e W ) = +oo )
3635adantr 481 . . . . . 6 |- ( ( ph /\ inf ( B , RR* , < ) = +oo ) -> ( +oo +e W ) = +oo )
3729, 36eqtr2d 2657 . . . . 5 |- ( ( ph /\ inf ( B , RR* , < ) = +oo ) -> +oo = (inf ( B , RR* , < ) +e W ) )
3827, 37breqtrd 4679 . . . 4 |- ( ( ph /\ inf ( B , RR* , < ) = +oo ) -> ( X +e ( W / 2 ) ) <_ (inf ( B , RR* , < ) +e W ) )
398, 17sseldd 3604 . . . . . . 7 |- ( ph -> X e. RR* )
4010, 22xaddcld 12131 . . . . . . 7 |- ( ph -> (inf ( B , RR* , < ) +e ( W / 2 ) ) e. RR* )
41 rphalfcl 11858 . . . . . . . . 9 |- ( W e. RR+ -> ( W / 2 ) e. RR+ )
4211, 41syl 17 . . . . . . . 8 |- ( ph -> ( W / 2 ) e. RR+ )
4342rpxrd 11873 . . . . . . 7 |- ( ph -> ( W / 2 ) e. RR* )
44 infleinflem1.i . . . . . . 7 |- ( ph -> X <_ (inf ( B , RR* , < ) +e ( W / 2 ) ) )
4539, 40, 43, 44xleadd1d 39545 . . . . . 6 |- ( ph -> ( X +e ( W / 2 ) ) <_ ( (inf ( B , RR* , < ) +e ( W / 2 ) ) +e ( W / 2 ) ) )
4645adantr 481 . . . . 5 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( X +e ( W / 2 ) ) <_ ( (inf ( B , RR* , < ) +e ( W / 2 ) ) +e ( W / 2 ) ) )
4710adantr 481 . . . . . . 7 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> inf ( B , RR* , < ) e. RR* )
48 neqne 2802 . . . . . . . 8 |- ( -. inf ( B , RR* , < ) = +oo -> inf ( B , RR* , < ) =/= +oo )
4948adantl 482 . . . . . . 7 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> inf ( B , RR* , < ) =/= +oo )
5043adantr 481 . . . . . . 7 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( W / 2 ) e. RR* )
5111adantr 481 . . . . . . . 8 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> W e. RR+ )
52 rpre 11839 . . . . . . . . 9 |- ( ( W / 2 ) e. RR+ -> ( W / 2 ) e. RR )
53 renepnf 10087 . . . . . . . . 9 |- ( ( W / 2 ) e. RR -> ( W / 2 ) =/= +oo )
5441, 52, 533syl 18 . . . . . . . 8 |- ( W e. RR+ -> ( W / 2 ) =/= +oo )
5551, 54syl 17 . . . . . . 7 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( W / 2 ) =/= +oo )
56 xaddass2 12080 . . . . . . 7 |- ( ( (inf ( B , RR* , < ) e. RR* /\ inf ( B , RR* , < ) =/= +oo ) /\ ( ( W / 2 ) e. RR* /\ ( W / 2 ) =/= +oo ) /\ ( ( W / 2 ) e. RR* /\ ( W / 2 ) =/= +oo ) ) -> ( (inf ( B , RR* , < ) +e ( W / 2 ) ) +e ( W / 2 ) ) = (inf ( B , RR* , < ) +e ( ( W / 2 ) +e ( W / 2 ) ) ) )
5747, 49, 50, 55, 50, 55, 56syl222anc 1342 . . . . . 6 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( (inf ( B , RR* , < ) +e ( W / 2 ) ) +e ( W / 2 ) ) = (inf ( B , RR* , < ) +e ( ( W / 2 ) +e ( W / 2 ) ) ) )
58 rehalfcl 11258 . . . . . . . . . 10 |- ( W e. RR -> ( W / 2 ) e. RR )
5958, 58rexaddd 12065 . . . . . . . . 9 |- ( W e. RR -> ( ( W / 2 ) +e ( W / 2 ) ) = ( ( W / 2 ) + ( W / 2 ) ) )
60 recn 10026 . . . . . . . . . 10 |- ( W e. RR -> W e. CC )
61 2halves 11260 . . . . . . . . . 10 |- ( W e. CC -> ( ( W / 2 ) + ( W / 2 ) ) = W )
6260, 61syl 17 . . . . . . . . 9 |- ( W e. RR -> ( ( W / 2 ) + ( W / 2 ) ) = W )
6359, 62eqtrd 2656 . . . . . . . 8 |- ( W e. RR -> ( ( W / 2 ) +e ( W / 2 ) ) = W )
6463oveq2d 6666 . . . . . . 7 |- ( W e. RR -> (inf ( B , RR* , < ) +e ( ( W / 2 ) +e ( W / 2 ) ) ) = (inf ( B , RR* , < ) +e W ) )
6551, 30, 643syl 18 . . . . . 6 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> (inf ( B , RR* , < ) +e ( ( W / 2 ) +e ( W / 2 ) ) ) = (inf ( B , RR* , < ) +e W ) )
6657, 65eqtrd 2656 . . . . 5 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( (inf ( B , RR* , < ) +e ( W / 2 ) ) +e ( W / 2 ) ) = (inf ( B , RR* , < ) +e W ) )
6746, 66breqtrd 4679 . . . 4 |- ( ( ph /\ -. inf ( B , RR* , < ) = +oo ) -> ( X +e ( W / 2 ) ) <_ (inf ( B , RR* , < ) +e W ) )
6838, 67pm2.61dan 832 . . 3 |- ( ph -> ( X +e ( W / 2 ) ) <_ (inf ( B , RR* , < ) +e W ) )
697, 23, 14, 24, 68xrletrd 11993 . 2 |- ( ph -> Z <_ (inf ( B , RR* , < ) +e W ) )
705, 7, 14, 16, 69xrletrd 11993 1 |- ( ph -> inf ( A , RR* , < ) <_ (inf ( B , RR* , < ) +e W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   RR+crp 11832   +ecxad 11944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947
This theorem is referenced by:  infleinf  39588
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