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Theorem sgplpte21 26132
 Description: The predicate "is a non-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
sgplpte21.2 btw
sgplpte21.6
sgplpte21.7
Assertion
Ref Expression
sgplpte21
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem sgplpte21
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgplpte.3 . . . 4
21a1i 10 . . 3
32oveqd 5875 . 2
4 df-seg2 26131 . . . . . . 7 Ibg PPoints PPoints PPoints btw
54a1i 10 . . . . . 6 Ibg PPoints PPoints PPoints btw
6 fveq2 5525 . . . . . . . 8 PPoints PPoints
76adantl 452 . . . . . . 7 PPoints PPoints
8 fveq2 5525 . . . . . . . . . . . . 13 btw btw
98adantl 452 . . . . . . . . . . . 12 btw btw
109oveqd 5875 . . . . . . . . . . 11 btw btw
1110eleq2d 2350 . . . . . . . . . 10 btw btw
12 biidd 228 . . . . . . . . . 10
13 biidd 228 . . . . . . . . . 10
1411, 12, 133orbi123d 1251 . . . . . . . . 9 btw btw
157, 14rabeqbidv 2783 . . . . . . . 8 PPoints btw PPoints btw
1615ifeq1d 3579 . . . . . . 7 PPoints btw PPoints btw
177, 7, 16mpt2eq123dv 5910 . . . . . 6 PPoints PPoints PPoints btw PPoints PPoints PPoints btw
18 sgplpte.4 . . . . . 6 Ibg
19 fvex 5539 . . . . . . . 8 PPoints
2019, 19mpt2ex 6198 . . . . . . 7 PPoints PPoints PPoints btw
2120a1i 10 . . . . . 6 PPoints PPoints PPoints btw
225, 17, 18, 21fvmptd 5606 . . . . 5 PPoints PPoints PPoints btw
2322oveqd 5875 . . . 4 PPoints PPoints PPoints btw
24 sgplpte.1 . . . . . . . 8 PPoints
2524eqcomi 2287 . . . . . . 7 PPoints
2625a1i 10 . . . . . 6 PPoints
27 sgplpte21.2 . . . . . . . . . . . . 13 btw
2827eqcomi 2287 . . . . . . . . . . . 12 btw
2928a1i 10 . . . . . . . . . . 11 btw
3029oveqd 5875 . . . . . . . . . 10 btw
3130eleq2d 2350 . . . . . . . . 9 btw
32 biidd 228 . . . . . . . . 9
33 biidd 228 . . . . . . . . 9
3431, 32, 333orbi123d 1251 . . . . . . . 8 btw
3526, 34rabeqbidv 2783 . . . . . . 7 PPoints btw
3635ifeq1d 3579 . . . . . 6 PPoints btw
3726, 26, 36mpt2eq123dv 5910 . . . . 5 PPoints PPoints PPoints btw
38 simprl 732 . . . . . . 7
39 simprr 733 . . . . . . 7
4038, 39neeq12d 2461 . . . . . 6
41 oveq12 5867 . . . . . . . . . 10
4241eleq2d 2350 . . . . . . . . 9
43 simpl 443 . . . . . . . . . 10
4443eqeq2d 2294 . . . . . . . . 9
45 simpr 447 . . . . . . . . . 10
4645eqeq2d 2294 . . . . . . . . 9
4742, 44, 463orbi123d 1251 . . . . . . . 8
4847adantl 452 . . . . . . 7
4948rabbidv 2780 . . . . . 6
50 sneq 3651 . . . . . . 7
5150ad2antrl 708 . . . . . 6
5240, 49, 51ifbieq12d 3587 . . . . 5
53 sgplpte.5 . . . . 5
54 sgplpte21.6 . . . . 5
5524, 19eqeltri 2353 . . . . . . . 8
5655rabex 4165 . . . . . . 7
57 snex 4216 . . . . . . 7
5856, 57ifex 3623 . . . . . 6
5958a1i 10 . . . . 5
6037, 52, 53, 54, 59ovmpt2d 5975 . . . 4 PPoints PPoints PPoints btw
6123, 60eqtrd 2315 . . 3
62 sgplpte21.7 . . . 4
63 iftrue 3571 . . . 4
6462, 63syl 15 . . 3
6561, 64eqtrd 2315 . 2
663, 65eqtrd 2315 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3o 933   wceq 1623   wcel 1684   wne 2446  crab 2547  cvv 2788  cif 3565  csn 3640   cmpt 4077  cfv 5255  (class class class)co 5858   cmpt2 5860  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107  cseg 26130 This theorem is referenced by:  sgplpte21a  26133  xsyysx  26145  bsstrs  26146  nbssntrs  26147 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-seg2 26131
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