Theorem lelttr 8268

Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)

Assertion

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

Proof of Theorem lelttr

Step	Hyp	Ref	Expression
1		simprl 531	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2		simpl1 1026	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A e. RR )
3		simpl2 1027	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B e. RR )
4		lenlt 8255	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
5	2, 3, 4	syl2anc 411	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( A <_ B <-> -. B < A ) )
6	1, 5	mpbid 147	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> -. B < A )
7	6	pm2.21d 624	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( B < A -> A < C ) )
8		idd 21	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( A < C -> A < C ) )
9		simprr 533	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10		simpl3 1028	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> C e. RR )
11		axltwlin 8247	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( B < C -> ( B < A \/ A < C ) ) )
12	3, 10, 2, 11	syl3anc 1273	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( B < C -> ( B < A \/ A < C ) ) )
13	9, 12	mpd 13	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( B < A \/ A < C ) )
14	7, 8, 13	mpjaod 725	. 2 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A < C )
15	14	ex 115	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   \/ wo 715   /\ w3a 1004   e. wcel 2202   class class class wbr 4088   RRcr 8031   < clt 8214   <_ cle 8215

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 8123  ax-resscn 8124  ax-pre-ltwlin 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 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220

This theorem is referenced by:  lelttri  8285  lelttrd  8304  letrp1  9028  ltmul12a  9040  bndndx  9401  uzind  9591  fnn0ind  9596  nn0p1elfzo  10422  elfzo0z  10424  fzofzim  10428  elfzodifsumelfzo  10447  flqge  10543  modfzo0difsn  10658  expnlbnd2  10928  ccat2s1fvwd  11228  swrdswrd  11290  pfxccatin12lem3  11317  caubnd2  11695  mulcn2  11890  cn1lem  11892  climsqz  11913  climsqz2  11914  climcvg1nlem  11927  ltoddhalfle  12472  algcvgblem  12639  pclemub  12878  metss2lem  15240  logdivlti  15624  gausslemma2dlem2  15810