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Theorem sgplpte22 26241
 Description: The predicate "is a degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
sgplpte.1 PPoints
sgplpte.3
sgplpte.4 Ibg
sgplpte.5
Assertion
Ref Expression
sgplpte22

Proof of Theorem sgplpte22
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgplpte.3 . . . 4
2 df-seg2 26234 . . . . . 6 Ibg PPoints PPoints PPoints btw
32a1i 10 . . . . 5 Ibg PPoints PPoints PPoints btw
4 fveq2 5541 . . . . . . 7 PPoints PPoints
54adantl 452 . . . . . 6 PPoints PPoints
6 fveq2 5541 . . . . . . . . . . . 12 btw btw
76adantl 452 . . . . . . . . . . 11 btw btw
87oveqd 5891 . . . . . . . . . 10 btw btw
98eleq2d 2363 . . . . . . . . 9 btw btw
10 biidd 228 . . . . . . . . 9
11 biidd 228 . . . . . . . . 9
129, 10, 113orbi123d 1251 . . . . . . . 8 btw btw
135, 12rabeqbidv 2796 . . . . . . 7 PPoints btw PPoints btw
1413ifeq1d 3592 . . . . . 6 PPoints btw PPoints btw
155, 5, 14mpt2eq123dv 5926 . . . . 5 PPoints PPoints PPoints btw PPoints PPoints PPoints btw
16 sgplpte.4 . . . . 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 PPoints PPoints PPoints btw
211, 20syl5eq 2340 . . 3 PPoints PPoints PPoints btw
2221oveqd 5891 . 2 PPoints PPoints PPoints btw
23 sgplpte.1 . . . . . 6 PPoints
2423eqcomi 2300 . . . . 5 PPoints
2524a1i 10 . . . 4 PPoints
26 biidd 228 . . . . . 6 btw btw
2725, 26rabeqbidv 2796 . . . . 5 PPoints btw btw
2827ifeq1d 3592 . . . 4 PPoints btw btw
2925, 25, 28mpt2eq123dv 5926 . . 3 PPoints PPoints PPoints btw btw
30 simpl 443 . . . . . 6
31 simpr 447 . . . . . 6
3230, 31neeq12d 2474 . . . . 5
3332adantl 452 . . . 4
34 oveq12 5883 . . . . . . . 8 btw btw
3534eleq2d 2363 . . . . . . 7 btw btw
3635adantl 452 . . . . . 6 btw btw
37 eqeq2 2305 . . . . . . 7
3837ad2antrl 708 . . . . . 6
39 eqeq2 2305 . . . . . . . 8
4039adantl 452 . . . . . . 7
4140adantl 452 . . . . . 6
4236, 38, 413orbi123d 1251 . . . . 5 btw btw
4342rabbidv 2793 . . . 4 btw btw
44 sneq 3664 . . . . 5
4544ad2antrl 708 . . . 4
4633, 43, 45ifbieq12d 3600 . . 3 btw btw
47 sgplpte.5 . . 3
4823, 17eqeltri 2366 . . . . . 6
4948rabex 4181 . . . . 5 btw
50 snex 4232 . . . . 5
5149, 50ifex 3636 . . . 4 btw
5251a1i 10 . . 3 btw
5329, 46, 47, 47, 52ovmpt2d 5991 . 2 PPoints PPoints PPoints btw btw
54 neirr 2464 . . 3
55 iffalse 3585 . . 3 btw
5654, 55mp1i 11 . 2 btw
5722, 53, 563eqtrd 2332 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   w3o 933   wceq 1632   wcel 1696   wne 2459  crab 2560  cvv 2801  cif 3578  csn 3653   cmpt 4093  cfv 5271  (class class class)co 5874   cmpt2 5876  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210  cseg 26233 This theorem is referenced by:  sgplpte21d1  26242  segline  26244  bsstrs  26249  nbssntrs  26250  segray  26258  bosser  26270  pdiveql  26271 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-3or 935  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-seg2 26234
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