Theorem ltleletr 8125

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 8117	. . . . . 6 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
2	1	3coml 1212	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
3	2	expcomd 1452	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C < A -> C < B ) ) )
4		con3 643	. . . 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 8119	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
7	6	3adant1 1017	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
8		lenlt 8119	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
9	8	3adant2 1018	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
10	7, 9	imbi12d 234	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B <_ C -> A <_ C ) <-> ( -. C < B -> -. C < A ) ) )
11	5, 10	sylibrd 169	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( B <_ C -> A <_ C ) ) )
12	11	impd 254	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2167   class class class wbr 4034   RRcr 7895   < clt 8078   <_ cle 8079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-lttrn 8010

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084

This theorem is referenced by:  nn0ge2m1nn  9326  lbzbi  9707  iseqf1olemqk  10616  wrdlenge2n0  10987  gausslemma2dlem3  15388  gausslemma2dlem4  15389