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Theorem qextltlem 12033
Description: Lemma for qextlt 12034 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
qextltlem |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( -. ( x < A <-> x < B ) /\ -. ( x <_ A <-> x <_ B ) ) ) )
Distinct variable groups:   x, A   x, B
Proof of Theorem qextltlem
StepHypRef Expression
1 qbtwnxr 12031 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
213expia 1267 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
3 simprl 794 . . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> A < x )
4 simplll 798 . . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> A e. RR* )
5 qre 11793 . . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
65rexrd 10089 . . . . . . . . . . 11 |- ( x e. QQ -> x e. RR* )
76ad2antlr 763 . . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> x e. RR* )
8 xrltnle 10105 . . . . . . . . . 10 |- ( ( A e. RR* /\ x e. RR* ) -> ( A < x <-> -. x <_ A ) )
94, 7, 8syl2anc 693 . . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( A < x <-> -. x <_ A ) )
103, 9mpbid 222 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> -. x <_ A )
11 xrltle 11982 . . . . . . . . 9 |- ( ( x e. RR* /\ A e. RR* ) -> ( x < A -> x <_ A ) )
127, 4, 11syl2anc 693 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( x < A -> x <_ A ) )
1310, 12mtod 189 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> -. x < A )
14 simprr 796 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> x < B )
1513, 142thd 255 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( -. x < A <-> x < B ) )
16 nbbn 373 . . . . . 6 |- ( ( -. x < A <-> x < B ) <-> -. ( x < A <-> x < B ) )
1715, 16sylib 208 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> -. ( x < A <-> x < B ) )
18 simpllr 799 . . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> B e. RR* )
19 xrltle 11982 . . . . . . . . 9 |- ( ( x e. RR* /\ B e. RR* ) -> ( x < B -> x <_ B ) )
207, 18, 19syl2anc 693 . . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( x < B -> x <_ B ) )
2114, 20mpd 15 . . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> x <_ B )
2210, 212thd 255 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( -. x <_ A <-> x <_ B ) )
23 nbbn 373 . . . . . 6 |- ( ( -. x <_ A <-> x <_ B ) <-> -. ( x <_ A <-> x <_ B ) )
2422, 23sylib 208 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> -. ( x <_ A <-> x <_ B ) )
2517, 24jca 554 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) /\ ( A < x /\ x < B ) ) -> ( -. ( x < A <-> x < B ) /\ -. ( x <_ A <-> x <_ B ) ) )
2625ex 450 . . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. QQ ) -> ( ( A < x /\ x < B ) -> ( -. ( x < A <-> x < B ) /\ -. ( x <_ A <-> x <_ B ) ) ) )
2726reximdva 3017 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. QQ ( -. ( x < A <-> x < B ) /\ -. ( x <_ A <-> x <_ B ) ) ) )
282, 27syld 47 1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( -. ( x < A <-> x < B ) /\ -. ( x <_ A <-> x <_ B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   class class class wbr 4653   RR*cxr 10073   < clt 10074   <_ cle 10075   QQcq 11788
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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789
This theorem is referenced by:  qextlt  12034  qextle  12035
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