Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isray2 Unicode version

Theorem isray2 26256
 Description: A degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 13-Apr-2016.)
Hypotheses
Ref Expression
isray2.1 PPoints
isray2.3 ray
isray2.5 Ibg
isray2.6
Assertion
Ref Expression
isray2

Proof of Theorem isray2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isray2.3 . . . 4 ray
2 df-ray2 26255 . . . . . 6 ray Ibg PPoints PPoints PPoints btw
32a1i 10 . . . . 5 ray Ibg PPoints PPoints PPoints btw
4 fveq2 5541 . . . . . . . 8 PPoints PPoints
5 isray2.1 . . . . . . . 8 PPoints
64, 5syl6eqr 2346 . . . . . . 7 PPoints
76adantl 452 . . . . . 6 PPoints
8 fveq2 5541 . . . . . . . . . 10
98oveqd 5891 . . . . . . . . 9
10 fveq2 5541 . . . . . . . . . . . 12 btw btw
1110oveqd 5891 . . . . . . . . . . 11 btw btw
1211eleq2d 2363 . . . . . . . . . 10 btw btw
134, 12rabeqbidv 2796 . . . . . . . . 9 PPoints btw PPoints btw
149, 13uneq12d 3343 . . . . . . . 8 PPoints btw PPoints btw
1514ifeq1d 3592 . . . . . . 7 PPoints btw PPoints btw
1615adantl 452 . . . . . 6 PPoints btw PPoints btw
177, 7, 16mpt2eq123dv 5926 . . . . 5 PPoints PPoints PPoints btw PPoints btw
18 isray2.5 . . . . 5 Ibg
19 fvex 5555 . . . . . . . 8 PPoints
205, 19eqeltri 2366 . . . . . . 7
2120, 20mpt2ex 6214 . . . . . 6 PPoints btw
2221a1i 10 . . . . 5 PPoints btw
233, 17, 18, 22fvmptd 5622 . . . 4 ray PPoints btw
241, 23syl5eq 2340 . . 3 PPoints btw
25 simpl 443 . . . . . 6
26 simpr 447 . . . . . 6
2725, 26neeq12d 2474 . . . . 5
28 oveq12 5883 . . . . . 6
29 oveq1 5881 . . . . . . . . 9 btw btw
3029adantr 451 . . . . . . . 8 btw btw
3126, 30eleq12d 2364 . . . . . . 7 btw btw
3231rabbidv 2793 . . . . . 6 PPoints btw PPoints btw
3328, 32uneq12d 3343 . . . . 5 PPoints btw PPoints btw
34 sneq 3664 . . . . . 6
3534adantr 451 . . . . 5
3627, 33, 35ifbieq12d 3600 . . . 4 PPoints btw PPoints btw
3736adantl 452 . . 3 PPoints btw PPoints btw
38 isray2.6 . . 3
39 ovex 5899 . . . . . 6
4019rabex 4181 . . . . . 6 PPoints btw
4139, 40unex 4534 . . . . 5 PPoints btw
42 snex 4232 . . . . 5
4341, 42ifex 3636 . . . 4 PPoints btw
4443a1i 10 . . 3 PPoints btw
4524, 37, 38, 38, 44ovmpt2d 5991 . 2 PPoints btw
46 neirr 2464 . . 3
47 iffalse 3585 . . 3 PPoints btw
4846, 47mp1i 11 . 2 PPoints btw
4945, 48eqtrd 2328 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wceq 1632   wcel 1696   wne 2459  crab 2560  cvv 2801   cun 3163  cif 3578  csn 3653   cmpt 4093  cfv 5271  (class class class)co 5874   cmpt2 5876  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210  cseg 26233  raycray2 26254 This theorem is referenced by:  segray  26258  rayline  26259 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ray2 26255
 Copyright terms: Public domain W3C validator