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Theorem isray 26257
 Description: A non-degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isray.1 PPoints
isray.2
isray.3 ray
isray.5 Ibg
isray.6a
isray.6b
isray.4 btw
isray.7
Assertion
Ref Expression
isray
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem isray
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isray.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 . . . . . . 7 PPoints PPoints
54adantl 452 . . . . . 6 PPoints PPoints
6 fveq2 5541 . . . . . . . . . 10
76oveqd 5891 . . . . . . . . 9
8 fveq2 5541 . . . . . . . . . . . 12 btw btw
98oveqd 5891 . . . . . . . . . . 11 btw btw
109eleq2d 2363 . . . . . . . . . 10 btw btw
114, 10rabeqbidv 2796 . . . . . . . . 9 PPoints btw PPoints btw
127, 11uneq12d 3343 . . . . . . . 8 PPoints btw PPoints btw
1312ifeq1d 3592 . . . . . . 7 PPoints btw PPoints btw
1413adantl 452 . . . . . 6 PPoints btw PPoints btw
155, 5, 14mpt2eq123dv 5926 . . . . 5 PPoints PPoints PPoints btw PPoints PPoints PPoints btw
16 isray.5 . . . . 5 Ibg
17 fvex 5555 . . . . . . 7 PPoints
1817, 17mpt2ex 6214 . . . . . 6 PPoints PPoints PPoints btw
1918a1i 10 . . . . 5 PPoints PPoints PPoints btw
203, 15, 16, 19fvmptd 5622 . . . 4 ray PPoints PPoints PPoints btw
211, 20syl5eq 2340 . . 3 PPoints PPoints PPoints btw
22 simpl 443 . . . . . 6
23 simpr 447 . . . . . 6
2422, 23neeq12d 2474 . . . . 5
25 isray.2 . . . . . . . . 9
2625eqcomi 2300 . . . . . . . 8
2726a1i 10 . . . . . . 7
2827, 22, 23oveq123d 5895 . . . . . 6
29 isray.1 . . . . . . . . 9 PPoints
3029eqcomi 2300 . . . . . . . 8 PPoints
3130a1i 10 . . . . . . 7 PPoints
32 isray.4 . . . . . . . . . . 11 btw
3332eqcomi 2300 . . . . . . . . . 10 btw
3433a1i 10 . . . . . . . . 9 btw
35 eqidd 2297 . . . . . . . . 9
3634, 22, 35oveq123d 5895 . . . . . . . 8 btw
3723, 36eleq12d 2364 . . . . . . 7 btw
3831, 37rabeqbidv 2796 . . . . . 6 PPoints btw
3928, 38uneq12d 3343 . . . . 5 PPoints btw
40 sneq 3664 . . . . . 6
4140adantr 451 . . . . 5
4224, 39, 41ifbieq12d 3600 . . . 4 PPoints btw
4342adantl 452 . . 3 PPoints btw
44 isray.6a . . . 4
4544, 29syl6eleq 2386 . . 3 PPoints
46 isray.6b . . . 4
4746, 29syl6eleq 2386 . . 3 PPoints
48 ovex 5899 . . . . . 6
4929, 17eqeltri 2366 . . . . . . 7
5049rabex 4181 . . . . . 6
5148, 50unex 4534 . . . . 5
52 snex 4232 . . . . 5
5351, 52ifex 3636 . . . 4
5453a1i 10 . . 3
5521, 43, 45, 47, 54ovmpt2d 5991 . 2
56 isray.7 . . 3
57 iftrue 3584 . . 3
5856, 57syl 15 . 2
5955, 58eqtrd 2328 1
 Colors of variables: wff set class Syntax hints:   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
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