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Theorem isside0 26267
 Description: The predicate "Being on the same side of " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
Hypotheses
Ref Expression
isside.1 PPoints
isside.2 PLines
isside.3 ss
isside.4 Ibg
isside.5
isside.7
Assertion
Ref Expression
isside0
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem isside0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isside.3 . 2 ss
2 isside.4 . . . . 5 Ibg
3 fvex 5555 . . . . . 6 PLines
43mptex 5762 . . . . 5 PLines PPoints PPoints
5 fveq2 5541 . . . . . . 7 PLines PLines
6 fveq2 5541 . . . . . . . . . . 11 PPoints PPoints
76difeq1d 3306 . . . . . . . . . 10 PPoints PPoints
87eleq2d 2363 . . . . . . . . 9 PPoints PPoints
97eleq2d 2363 . . . . . . . . 9 PPoints PPoints
10 fveq2 5541 . . . . . . . . . . . 12
1110oveqd 5891 . . . . . . . . . . 11
1211ineq1d 3382 . . . . . . . . . 10
1312eqeq1d 2304 . . . . . . . . 9
148, 9, 133anbi123d 1252 . . . . . . . 8 PPoints PPoints PPoints PPoints
1514opabbidv 4098 . . . . . . 7 PPoints PPoints PPoints PPoints
165, 15mpteq12dv 4114 . . . . . 6 PLines PPoints PPoints PLines PPoints PPoints
17 df-sside 26266 . . . . . 6 ss Ibg PLines PPoints PPoints
1816, 17fvmptg 5616 . . . . 5 Ibg PLines PPoints PPoints ss PLines PPoints PPoints
192, 4, 18sylancl 643 . . . 4 ss PLines PPoints PPoints
2019fveq1d 5543 . . 3 ss PLines PPoints PPoints
21 isside.5 . . . . 5
22 isside.2 . . . . 5 PLines
2321, 22syl6eleq 2386 . . . 4 PLines
24 difeq2 3301 . . . . . . . 8 PPoints PPoints
2524eleq2d 2363 . . . . . . 7 PPoints PPoints
2624eleq2d 2363 . . . . . . 7 PPoints PPoints
27 ineq2 3377 . . . . . . . 8
2827eqeq1d 2304 . . . . . . 7
2925, 26, 283anbi123d 1252 . . . . . 6 PPoints PPoints PPoints PPoints
3029opabbidv 4098 . . . . 5 PPoints PPoints PPoints PPoints
31 eqid 2296 . . . . 5 PLines PPoints PPoints PLines PPoints PPoints
32 3anass 938 . . . . . . . 8 PPoints PPoints PPoints PPoints
3332opabbii 4099 . . . . . . 7 PPoints PPoints PPoints PPoints
3433a1i 10 . . . . . 6 PLines PPoints PPoints PPoints PPoints
35 fvex 5555 . . . . . . . 8 PPoints
36 difexg 4178 . . . . . . . 8 PPoints PPoints
3735, 36ax-mp 8 . . . . . . 7 PPoints
3837zfausab 4179 . . . . . . . 8 PPoints
3938a1i 10 . . . . . . 7 PPoints PPoints
4037, 39opabex3 5785 . . . . . 6 PPoints PPoints
4134, 40syl6eqel 2384 . . . . 5 PLines PPoints PPoints
4230, 31, 41fvmpt3 5620 . . . 4 PLines PLines PPoints PPoints PPoints PPoints
4323, 42syl 15 . . 3 PLines PPoints PPoints PPoints PPoints
44 isside.1 . . . . . . . . 9 PPoints
4544eqcomi 2300 . . . . . . . 8 PPoints
4645difeq1i 3303 . . . . . . 7 PPoints
4746a1i 10 . . . . . 6 PPoints
4847eleq2d 2363 . . . . 5 PPoints
4947eleq2d 2363 . . . . 5 PPoints
50 isside.7 . . . . . . . . . 10
5150eqcomi 2300 . . . . . . . . 9
5251a1i 10 . . . . . . . 8
5352oveqd 5891 . . . . . . 7
5453ineq1d 3382 . . . . . 6
5554eqeq1d 2304 . . . . 5
5648, 49, 553anbi123d 1252 . . . 4 PPoints PPoints
5756opabbidv 4098 . . 3 PPoints PPoints
5820, 43, 573eqtrd 2332 . 2 ss
591, 58syl5eq 2340 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696  cab 2282  cvv 2801   cdif 3162   cin 3164  c0 3468  copab 4092   cmpt 4093  cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Ibgcibg 26210  cseg 26233  sscsas 26265 This theorem is referenced by:  isside1  26268  bosser  26270 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-sside 26266
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