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Theorem infleinflem2 37681
Description: Lemma for infleinf 37682, when inf ( B ,  RR* ,  <  )  = -oo. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
infleinflem2.a  |-  ( ph  ->  A  C_  RR* )
infleinflem2.b  |-  ( ph  ->  B  C_  RR* )
infleinflem2.r  |-  ( ph  ->  R  e.  RR )
infleinflem2.x  |-  ( ph  ->  X  e.  B )
infleinflem2.t  |-  ( ph  ->  X  <  ( R  -  2 ) )
infleinflem2.z  |-  ( ph  ->  Z  e.  A )
infleinflem2.l  |-  ( ph  ->  Z  <_  ( X +e 1 ) )
Assertion
Ref Expression
infleinflem2  |-  ( ph  ->  Z  <  R )

Proof of Theorem infleinflem2
StepHypRef Expression
1 infleinflem2.r . . . 4  |-  ( ph  ->  R  e.  RR )
21adantr 472 . . 3  |-  ( (
ph  /\  Z  = -oo )  ->  R  e.  RR )
3 simpr 468 . . 3  |-  ( (
ph  /\  Z  = -oo )  ->  Z  = -oo )
4 simpr 468 . . . 4  |-  ( ( R  e.  RR  /\  Z  = -oo )  ->  Z  = -oo )
5 mnflt 11448 . . . . 5  |-  ( R  e.  RR  -> -oo  <  R )
65adantr 472 . . . 4  |-  ( ( R  e.  RR  /\  Z  = -oo )  -> -oo  <  R )
74, 6eqbrtrd 4416 . . 3  |-  ( ( R  e.  RR  /\  Z  = -oo )  ->  Z  <  R )
82, 3, 7syl2anc 673 . 2  |-  ( (
ph  /\  Z  = -oo )  ->  Z  < 
R )
9 simpl 464 . . 3  |-  ( (
ph  /\  -.  Z  = -oo )  ->  ph )
10 neqne 2651 . . . 4  |-  ( -.  Z  = -oo  ->  Z  =/= -oo )
1110adantl 473 . . 3  |-  ( (
ph  /\  -.  Z  = -oo )  ->  Z  =/= -oo )
121adantr 472 . . . . 5  |-  ( (
ph  /\  Z  =/= -oo )  ->  R  e.  RR )
13 id 22 . . . . . . . 8  |-  ( ph  ->  ph )
14 infleinflem2.x . . . . . . . 8  |-  ( ph  ->  X  e.  B )
15 infleinflem2.b . . . . . . . . 9  |-  ( ph  ->  B  C_  RR* )
1615sselda 3418 . . . . . . . 8  |-  ( (
ph  /\  X  e.  B )  ->  X  e.  RR* )
1713, 14, 16syl2anc 673 . . . . . . 7  |-  ( ph  ->  X  e.  RR* )
1817adantr 472 . . . . . 6  |-  ( (
ph  /\  Z  =/= -oo )  ->  X  e.  RR* )
19 infleinflem2.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  A )
20 infleinflem2.a . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR* )
2120sselda 3418 . . . . . . . . . 10  |-  ( (
ph  /\  Z  e.  A )  ->  Z  e.  RR* )
2213, 19, 21syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  Z  e.  RR* )
2322adantr 472 . . . . . . . 8  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  e.  RR* )
24 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  =/= -oo )
25 pnfxr 11435 . . . . . . . . . . 11  |- +oo  e.  RR*
2625a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
27 peano2rem 9961 . . . . . . . . . . . . 13  |-  ( R  e.  RR  ->  ( R  -  1 )  e.  RR )
2827rexrd 9708 . . . . . . . . . . . 12  |-  ( R  e.  RR  ->  ( R  -  1 )  e.  RR* )
291, 28syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( R  -  1 )  e.  RR* )
3015, 14sseldd 3419 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  RR* )
31 id 22 . . . . . . . . . . . . . 14  |-  ( X  e.  RR*  ->  X  e. 
RR* )
32 1re 9660 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
3332rexri 9711 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
3433a1i 11 . . . . . . . . . . . . . 14  |-  ( X  e.  RR*  ->  1  e. 
RR* )
3531, 34xaddcld 11612 . . . . . . . . . . . . 13  |-  ( X  e.  RR*  ->  ( X +e 1 )  e.  RR* )
3630, 35syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( X +e 1 )  e.  RR* )
37 infleinflem2.l . . . . . . . . . . . 12  |-  ( ph  ->  Z  <_  ( X +e 1 ) )
38 infleinflem2.t . . . . . . . . . . . . 13  |-  ( ph  ->  X  <  ( R  -  2 ) )
39 oveq1 6315 . . . . . . . . . . . . . . . . . . 19  |-  ( X  = -oo  ->  ( X +e 1 )  =  ( -oo +e 1 ) )
40 renepnf 9706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  e.  RR  ->  1  =/= +oo )
4132, 40ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  1  =/= +oo
42 xaddmnf2 11545 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  RR*  /\  1  =/= +oo )  ->  ( -oo +e 1 )  = -oo )
4333, 41, 42mp2an 686 . . . . . . . . . . . . . . . . . . . 20  |-  ( -oo +e 1 )  = -oo
4443a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( X  = -oo  ->  ( -oo +e 1 )  = -oo )
4539, 44eqtrd 2505 . . . . . . . . . . . . . . . . . 18  |-  ( X  = -oo  ->  ( X +e 1 )  = -oo )
4645adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RR  /\  X  = -oo )  ->  ( X +e 1 )  = -oo )
4727mnfltd 11449 . . . . . . . . . . . . . . . . . 18  |-  ( R  e.  RR  -> -oo  <  ( R  -  1 ) )
4847adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RR  /\  X  = -oo )  -> -oo  <  ( R  -  1 ) )
4946, 48eqbrtrd 4416 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  RR  /\  X  = -oo )  ->  ( X +e 1 )  <  ( R  -  1 ) )
5049adantlr 729 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RR  /\  X  e.  RR* )  /\  X  = -oo )  ->  ( X +e 1 )  < 
( R  -  1 ) )
51503adantl3 1188 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  X  = -oo )  ->  ( X +e 1 )  <  ( R  - 
1 ) )
52 simpl 464 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) ) )
53 simpl2 1034 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  X  e.  RR* )
54 neqne 2651 . . . . . . . . . . . . . . . . 17  |-  ( -.  X  = -oo  ->  X  =/= -oo )
5554adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  X  =/= -oo )
56 simp2 1031 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  X  e.  RR* )
5725a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  -> +oo  e.  RR* )
58 id 22 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  RR  ->  R  e.  RR )
59 2re 10701 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  RR
6059a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  RR  ->  2  e.  RR )
6158, 60resubcld 10068 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  e.  RR  ->  ( R  -  2 )  e.  RR )
6261rexrd 9708 . . . . . . . . . . . . . . . . . . . 20  |-  ( R  e.  RR  ->  ( R  -  2 )  e.  RR* )
63623ad2ant1 1051 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  ( R  -  2 )  e.  RR* )
64 simp3 1032 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  X  <  ( R  -  2 ) )
6561ltpnfd 11446 . . . . . . . . . . . . . . . . . . . 20  |-  ( R  e.  RR  ->  ( R  -  2 )  < +oo )
66653ad2ant1 1051 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  ( R  -  2 )  < +oo )
6756, 63, 57, 64, 66xrlttrd 11479 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  X  < +oo )
6856, 57, 67xrltned 37667 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  X  =/= +oo )
6968adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  X  =/= +oo )
7053, 55, 69xrred 37675 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  X  e.  RR )
71 id 22 . . . . . . . . . . . . . . . . . . . . 21  |-  ( X  e.  RR  ->  X  e.  RR )
7271ad2antlr 741 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  X  e.  RR )
7361ad2antrr 740 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( R  -  2 )  e.  RR )
74 1red 9676 . . . . . . . . . . . . . . . . . . . . 21  |-  ( X  e.  RR  ->  1  e.  RR )
7572, 74syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  1  e.  RR )
76 simpr 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  X  <  ( R  -  2 ) )
7772, 73, 75, 76ltadd1dd 10245 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( X  +  1 )  < 
( ( R  - 
2 )  +  1 ) )
78 recn 9647 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  e.  RR  ->  R  e.  CC )
79 id 22 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( R  e.  CC  ->  R  e.  CC )
80 2cnd 10704 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( R  e.  CC  ->  2  e.  CC )
81 1cnd 9677 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( R  e.  CC  ->  1  e.  CC )
8279, 80, 81subsubd 10033 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  CC  ->  ( R  -  ( 2  -  1 ) )  =  ( ( R  -  2 )  +  1 ) )
83 2m1e1 10746 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 2  -  1 )  =  1
8483oveq2i 6319 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( R  -  ( 2  -  1 ) )  =  ( R  -  1 )
8584a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  CC  ->  ( R  -  ( 2  -  1 ) )  =  ( R  - 
1 ) )
8682, 85eqtr3d 2507 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  e.  CC  ->  (
( R  -  2 )  +  1 )  =  ( R  - 
1 ) )
8778, 86syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( R  e.  RR  ->  (
( R  -  2 )  +  1 )  =  ( R  - 
1 ) )
8887ad2antrr 740 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( ( R  -  2 )  +  1 )  =  ( R  -  1 ) )
8977, 88breqtrd 4420 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( X  +  1 )  < 
( R  -  1 ) )
9071, 74rexaddd 11550 . . . . . . . . . . . . . . . . . . . 20  |-  ( X  e.  RR  ->  ( X +e 1 )  =  ( X  + 
1 ) )
9190breq1d 4405 . . . . . . . . . . . . . . . . . . 19  |-  ( X  e.  RR  ->  (
( X +e 1 )  <  ( R  -  1 )  <-> 
( X  +  1 )  <  ( R  -  1 ) ) )
9291ad2antlr 741 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( ( X +e 1 )  <  ( R  - 
1 )  <->  ( X  +  1 )  < 
( R  -  1 ) ) )
9389, 92mpbird 240 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  ->  ( X +e 1 )  <  ( R  - 
1 ) )
9493an32s 821 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RR  /\  X  <  ( R  -  2 ) )  /\  X  e.  RR )  ->  ( X +e 1 )  < 
( R  -  1 ) )
95943adantl2 1187 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  X  e.  RR )  ->  ( X +e 1 )  <  ( R  - 
1 ) )
9652, 70, 95syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  /\  -.  X  = -oo )  ->  ( X +e 1 )  <  ( R  -  1 ) )
9751, 96pm2.61dan 808 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  X  e.  RR*  /\  X  <  ( R  -  2 ) )  ->  ( X +e 1 )  <  ( R  - 
1 ) )
981, 30, 38, 97syl3anc 1292 . . . . . . . . . . . 12  |-  ( ph  ->  ( X +e 1 )  <  ( R  -  1 ) )
9922, 36, 29, 37, 98xrlelttrd 11480 . . . . . . . . . . 11  |-  ( ph  ->  Z  <  ( R  -  1 ) )
10027ltpnfd 11446 . . . . . . . . . . . 12  |-  ( R  e.  RR  ->  ( R  -  1 )  < +oo )
1011, 100syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( R  -  1 )  < +oo )
10222, 29, 26, 99, 101xrlttrd 11479 . . . . . . . . . 10  |-  ( ph  ->  Z  < +oo )
10322, 26, 102xrltned 37667 . . . . . . . . 9  |-  ( ph  ->  Z  =/= +oo )
104103adantr 472 . . . . . . . 8  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  =/= +oo )
10523, 24, 104xrred 37675 . . . . . . 7  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  e.  RR )
10637adantr 472 . . . . . . 7  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  <_  ( X +e 1 ) )
107 simpl3 1035 . . . . . . . . 9  |-  ( ( ( Z  e.  RR  /\  X  e.  RR*  /\  Z  <_  ( X +e 1 ) )  /\  X  = -oo )  ->  Z  <_  ( X +e 1 ) )
10845adantl 473 . . . . . . . . . . . 12  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  ( X +e 1 )  = -oo )
109 mnflt 11448 . . . . . . . . . . . . 13  |-  ( Z  e.  RR  -> -oo  <  Z )
110109adantr 472 . . . . . . . . . . . 12  |-  ( ( Z  e.  RR  /\  X  = -oo )  -> -oo  <  Z )
111108, 110eqbrtrd 4416 . . . . . . . . . . 11  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  ( X +e 1 )  <  Z
)
112 mnfxr 11437 . . . . . . . . . . . . 13  |- -oo  e.  RR*
113108, 112syl6eqel 2557 . . . . . . . . . . . 12  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  ( X +e 1 )  e.  RR* )
114 rexr 9704 . . . . . . . . . . . . 13  |-  ( Z  e.  RR  ->  Z  e.  RR* )
115114adantr 472 . . . . . . . . . . . 12  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  Z  e.  RR* )
116113, 115xrltnled 37673 . . . . . . . . . . 11  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  ( ( X +e 1 )  < 
Z  <->  -.  Z  <_  ( X +e 1 ) ) )
117111, 116mpbid 215 . . . . . . . . . 10  |-  ( ( Z  e.  RR  /\  X  = -oo )  ->  -.  Z  <_  ( X +e 1 ) )
1181173ad2antl1 1192 . . . . . . . . 9  |-  ( ( ( Z  e.  RR  /\  X  e.  RR*  /\  Z  <_  ( X +e 1 ) )  /\  X  = -oo )  ->  -.  Z  <_  ( X +e 1 ) )
119107, 118pm2.65da 586 . . . . . . . 8  |-  ( ( Z  e.  RR  /\  X  e.  RR*  /\  Z  <_  ( X +e 1 ) )  ->  -.  X  = -oo )
120119neqned 2650 . . . . . . 7  |-  ( ( Z  e.  RR  /\  X  e.  RR*  /\  Z  <_  ( X +e 1 ) )  ->  X  =/= -oo )
121105, 18, 106, 120syl3anc 1292 . . . . . 6  |-  ( (
ph  /\  Z  =/= -oo )  ->  X  =/= -oo )
1221, 17, 38, 68syl3anc 1292 . . . . . . 7  |-  ( ph  ->  X  =/= +oo )
123122adantr 472 . . . . . 6  |-  ( (
ph  /\  Z  =/= -oo )  ->  X  =/= +oo )
12418, 121, 123xrred 37675 . . . . 5  |-  ( (
ph  /\  Z  =/= -oo )  ->  X  e.  RR )
12538adantr 472 . . . . 5  |-  ( (
ph  /\  Z  =/= -oo )  ->  X  <  ( R  -  2 ) )
12612, 124, 125jca31 543 . . . 4  |-  ( (
ph  /\  Z  =/= -oo )  ->  ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) ) )
127 simplr 770 . . . . 5  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  ->  Z  e.  RR )
128 simp-4r 785 . . . . . 6  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  ->  X  e.  RR )
12971, 74readdcld 9688 . . . . . . 7  |-  ( X  e.  RR  ->  ( X  +  1 )  e.  RR )
13090, 129eqeltrd 2549 . . . . . 6  |-  ( X  e.  RR  ->  ( X +e 1 )  e.  RR )
131128, 130syl 17 . . . . 5  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  -> 
( X +e 1 )  e.  RR )
13258ad4antr 746 . . . . 5  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  ->  R  e.  RR )
133 simpr 468 . . . . 5  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  ->  Z  <_  ( X +e 1 ) )
134130ad3antlr 745 . . . . . . 7  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  ( X +e 1 )  e.  RR )
13527ad3antrrr 744 . . . . . . 7  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  ( R  -  1 )  e.  RR )
13658ad3antrrr 744 . . . . . . 7  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  R  e.  RR )
13793adantr 472 . . . . . . 7  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  ( X +e 1 )  <  ( R  - 
1 ) )
138136ltm1d 10561 . . . . . . 7  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  ( R  -  1 )  <  R )
139134, 135, 136, 137, 138lttrd 9813 . . . . . 6  |-  ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  ->  ( X +e 1 )  <  R )
140139adantr 472 . . . . 5  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  -> 
( X +e 1 )  <  R
)
141127, 131, 132, 133, 140lelttrd 9810 . . . 4  |-  ( ( ( ( ( R  e.  RR  /\  X  e.  RR )  /\  X  <  ( R  -  2 ) )  /\  Z  e.  RR )  /\  Z  <_  ( X +e 1 ) )  ->  Z  <  R )
142126, 105, 106, 141syl21anc 1291 . . 3  |-  ( (
ph  /\  Z  =/= -oo )  ->  Z  <  R )
1439, 11, 142syl2anc 673 . 2  |-  ( (
ph  /\  -.  Z  = -oo )  ->  Z  <  R )
1448, 143pm2.61dan 808 1  |-  ( ph  ->  Z  <  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   class class class wbr 4395  (class class class)co 6308   CCcc 9555   RRcr 9556   1c1 9558    + caddc 9560   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   2c2 10681   +ecxad 11430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-2 10690  df-xadd 11433
This theorem is referenced by:  infleinf  37682
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