Theorem ltleletr 8260

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 8252	. . . . . 6 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
2	1	3coml 1236	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C < A /\ A < B ) -> C < B ) )
3	2	expcomd 1486	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C < A -> C < B ) ) )
4		con3 647	. . . 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 8254	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
7	6	3adant1 1041	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
8		lenlt 8254	. . . . 5 |- ( ( A e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
9	8	3adant2 1042	. . . 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 1004   e. wcel 2202   class class class wbr 4088   RRcr 8030   < clt 8213   <_ cle 8214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by:  nn0ge2m1nn  9461  lbzbi  9849  iseqf1olemqk  10768  wrdlenge2n0  11148  gausslemma2dlem3  15791  gausslemma2dlem4  15792