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Theorem aishp 26275
 Description: The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
abhp.1 Ibg
abhp.2
abhp.3 PPoints
abhp.4 ss
abhp.5 PLines
Assertion
Ref Expression
aishp Halfplane

Proof of Theorem aishp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-halfplane 26274 . . . . 5 Halfplane Ibg PLines PPoints ss
21a1i 10 . . . 4 Halfplane Ibg PLines PPoints ss
3 fveq2 5541 . . . . . 6 PLines PLines
4 fveq2 5541 . . . . . . . . 9 PPoints PPoints
54difeq1d 3306 . . . . . . . 8 PPoints PPoints
6 qseq1 6725 . . . . . . . 8 PPoints PPoints PPoints ss PPoints ss
75, 6syl 15 . . . . . . 7 PPoints ss PPoints ss
8 fveq2 5541 . . . . . . . . 9 ss ss
98fveq1d 5543 . . . . . . . 8 ss ss
10 qseq2 6726 . . . . . . . 8 ss ss PPoints ss PPoints ss
119, 10syl 15 . . . . . . 7 PPoints ss PPoints ss
127, 11eqtrd 2328 . . . . . 6 PPoints ss PPoints ss
133, 12mpteq12dv 4114 . . . . 5 PLines PPoints ss PLines PPoints ss
1413adantl 452 . . . 4 PLines PPoints ss PLines PPoints ss
15 abhp.1 . . . 4 Ibg
16 fvex 5555 . . . . 5 PLines
17 mptexg 5761 . . . . 5 PLines PLines PPoints ss
1816, 17mp1i 11 . . . 4 PLines PPoints ss
192, 14, 15, 18fvmptd 5622 . . 3 Halfplane PLines PPoints ss
20 difeq2 3301 . . . . . 6 PPoints PPoints
21 qseq1 6725 . . . . . 6 PPoints PPoints PPoints ss PPoints ss
2220, 21syl 15 . . . . 5 PPoints ss PPoints ss
23 fveq2 5541 . . . . . 6 ss ss
24 qseq2 6726 . . . . . 6 ss ss PPoints ss PPoints ss
2523, 24syl 15 . . . . 5 PPoints ss PPoints ss
2622, 25eqtrd 2328 . . . 4 PPoints ss PPoints ss
2726adantl 452 . . 3 PPoints ss PPoints ss
28 abhp.2 . . . 4
29 abhp.5 . . . 4 PLines
3028, 29syl6eleq 2386 . . 3 PLines
31 fvex 5555 . . . . . 6 PPoints
32 difexg 4178 . . . . . 6 PPoints PPoints
3331, 32ax-mp 8 . . . . 5 PPoints
3433qsex 6734 . . . 4 PPoints ss
3534a1i 10 . . 3 PPoints ss
3619, 27, 30, 35fvmptd 5622 . 2 Halfplane PPoints ss
37 abhp.3 . . . . 5 PPoints
3837eqcomi 2300 . . . 4 PPoints
3938difeq1i 3303 . . 3 PPoints
40 qseq1 6725 . . 3 PPoints PPoints ss ss
4139, 40mp1i 11 . 2 PPoints ss ss
42 abhp.4 . . . 4 ss
4342eqcomi 2300 . . 3 ss
44 qseq2 6726 . . 3 ss ss
4543, 44mp1i 11 . 2 ss
4636, 41, 453eqtrd 2332 1 Halfplane
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1632   wcel 1696  cvv 2801   cdif 3162   cmpt 4093  cfv 5271  cqs 6675  PPointscpoints 26159  PLinescplines 26161  Ibgcibg 26210  sscsas 26265  Halfplanechalfp 26273 This theorem is referenced by:  abhp  26276  abhp1  26277 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-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-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-ec 6678  df-qs 6682  df-halfplane 26274
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