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Theorem xrsdsreclblem 16736
Description: Lemma for xrsdsreclb 16737. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
xrsds.d  |-  D  =  ( dist `  RR* s
)
Assertion
Ref Expression
xrsdsreclblem  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  A  <_  B )  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )

Proof of Theorem xrsdsreclblem
StepHypRef Expression
1 necom 2679 . . . . 5  |-  ( A  =/=  B  <->  B  =/=  A )
2 xrleltne 10730 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  <  B  <->  B  =/=  A ) )
3 mnfxr 10706 . . . . . . . . . . . 12  |-  -oo  e.  RR*
43a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -oo  e.  RR* )
5 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  A  e.  RR* )
6 simpl2 961 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  B  e.  RR* )
7 pnfnre 9119 . . . . . . . . . . . . . 14  |-  +oo  e/  RR
8 df-nel 2601 . . . . . . . . . . . . . 14  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
97, 8mpbi 200 . . . . . . . . . . . . 13  |-  -.  +oo  e.  RR
10 mnfle 10721 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  RR*  ->  -oo  <_  A )
115, 10syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -oo  <_  A )
12 simpl3 962 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  A  <  B
)
134, 5, 6, 11, 12xrlelttrd 10742 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -oo  <  B )
14 xrltne 10745 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  -oo  <  B )  ->  B  =/=  -oo )
154, 6, 13, 14syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  B  =/=  -oo )
16 xaddpnf1 10804 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  ( B + e  +oo )  =  +oo )
176, 15, 16syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( B + e  +oo )  =  +oo )
1817eleq1d 2501 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( ( B + e  +oo )  e.  RR  <->  +oo  e.  RR ) )
199, 18mtbiri 295 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -.  ( B + e  +oo )  e.  RR )
20 ngtmnft 10747 . . . . . . . . . . . . . 14  |-  ( A  e.  RR*  ->  ( A  =  -oo  <->  -.  -oo  <  A ) )
215, 20syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( A  = 
-oo 
<->  -.  -oo  <  A ) )
22 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( B + e  - e A )  e.  RR )
23 xnegeq 10785 . . . . . . . . . . . . . . . . 17  |-  ( A  =  -oo  ->  - e A  =  - e  -oo )
24 xnegmnf 10788 . . . . . . . . . . . . . . . . 17  |-  - e  -oo  =  +oo
2523, 24syl6eq 2483 . . . . . . . . . . . . . . . 16  |-  ( A  =  -oo  ->  - e A  =  +oo )
2625oveq2d 6089 . . . . . . . . . . . . . . 15  |-  ( A  =  -oo  ->  ( B + e  - e A )  =  ( B + e  +oo ) )
2726eleq1d 2501 . . . . . . . . . . . . . 14  |-  ( A  =  -oo  ->  (
( B + e  - e A )  e.  RR  <->  ( B + e  +oo )  e.  RR ) )
2822, 27syl5ibcom 212 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( A  = 
-oo  ->  ( B + e  +oo )  e.  RR ) )
2921, 28sylbird 227 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( -.  -oo  <  A  ->  ( B + e  +oo )  e.  RR ) )
3019, 29mt3d 119 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -oo  <  A )
31 xrre2 10750 . . . . . . . . . . 11  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  (  -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
324, 5, 6, 30, 12, 31syl32anc 1192 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  A  e.  RR )
33 pnfxr 10705 . . . . . . . . . . . 12  |-  +oo  e.  RR*
3433a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  +oo  e.  RR* )
355xnegcld 10871 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  - e A  e. 
RR* )
36 xnegpnf 10787 . . . . . . . . . . . . . . . . 17  |-  - e  +oo  =  -oo
37 pnfge 10719 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  e.  RR*  ->  B  <_  +oo )
386, 37syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  B  <_  +oo )
395, 6, 34, 12, 38xrltletrd 10743 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  A  <  +oo )
40 xltnegi 10794 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR*  /\  +oo  e.  RR*  /\  A  <  +oo )  ->  - e  +oo  <  - e A )
415, 34, 39, 40syl3anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  - e  +oo  <  - e A )
4236, 41syl5eqbrr 4238 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -oo  <  - e A )
43 xrltne 10745 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  - e A  e.  RR*  /\  -oo  <  - e A )  ->  - e A  =/= 
-oo )
444, 35, 42, 43syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  - e A  =/= 
-oo )
45 xaddpnf2 10805 . . . . . . . . . . . . . . 15  |-  ( ( 
- e A  e. 
RR*  /\  - e A  =/=  -oo )  ->  (  +oo + e  - e A )  =  +oo )
4635, 44, 45syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  (  +oo + e  - e A )  =  +oo )
4746eleq1d 2501 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( (  +oo + e  - e A )  e.  RR  <->  +oo  e.  RR ) )
489, 47mtbiri 295 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  -.  (  +oo + e  - e A )  e.  RR )
49 nltpnft 10746 . . . . . . . . . . . . . 14  |-  ( B  e.  RR*  ->  ( B  =  +oo  <->  -.  B  <  +oo ) )
506, 49syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( B  = 
+oo 
<->  -.  B  <  +oo ) )
51 oveq1 6080 . . . . . . . . . . . . . . 15  |-  ( B  =  +oo  ->  ( B + e  - e A )  =  ( 
+oo + e  - e A ) )
5251eleq1d 2501 . . . . . . . . . . . . . 14  |-  ( B  =  +oo  ->  (
( B + e  - e A )  e.  RR  <->  (  +oo + e  - e A )  e.  RR ) )
5322, 52syl5ibcom 212 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( B  = 
+oo  ->  (  +oo + e  - e A )  e.  RR ) )
5450, 53sylbird 227 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( -.  B  <  +oo  ->  (  +oo + e  - e A )  e.  RR ) )
5548, 54mt3d 119 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  B  <  +oo )
56 xrre2 10750 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
575, 6, 34, 12, 55, 56syl32anc 1192 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  B  e.  RR )
5832, 57jca 519 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( B + e  - e A )  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR ) )
5958ex 424 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )
60593expia 1155 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
61603adant3 977 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  <  B  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
622, 61sylbird 227 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( B  =/=  A  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
631, 62syl5bi 209 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  =/=  B  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
64633exp 1152 . . 3  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( A  <_  B  ->  ( A  =/=  B  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) ) ) )
6564com34 79 . 2  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( A  =/=  B  ->  ( A  <_  B  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) ) ) )
66653imp1 1166 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  A  <_  B )  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113    - ecxne 10699   + ecxad 10700   distcds 13530   RR* scxrs 13714
This theorem is referenced by:  xrsdsreclb  16737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-xneg 10702  df-xadd 10703
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