Theorem ltleletr 7980

Description: Transitive law, weaker form of ( A < B /\ B <_ C ) -> A < C. (Contributed by AV, 14-Oct-2018.)

Assertion

Ref	Expression
ltleletr	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )

Proof of Theorem ltleletr

Step	Hyp	Ref	Expression
1		lttr 7972	. . . . . 6 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
2	1	3coml 1200	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
3	2	expcomd 1429	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C < A -> C < B ) ) )
4		con3 632	. . . 4 |- ( ( C < A -> C < B ) -> ( -. C < B -> -. C < A ) )
5	3, 4	syl6 33	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( -. C < B -> -. C < A ) ) )
6		lenlt 7974	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
7	6	3adant1 1005	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
8		lenlt 7974	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
9	8	3adant2 1006	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
10	7, 9	imbi12d 233	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B <_ C -> A <_ C ) <-> ( -. C < B -> -. C < A ) ) )
11	5, 10	sylibrd 168	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( B <_ C -> A <_ C ) ) )
12	11	impd 252	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   e. wcel 2136   class class class wbr 3982   RRcr 7752   < clt 7933   <_ cle 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-lttrn 7867

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by:  nn0ge2m1nn  9174  lbzbi  9554  iseqf1olemqk  10429