Theorem tglinecom 25530

Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	|- B = ( Base ` G )
tglineelsb2.i	|- I = (Itv ` G )
tglineelsb2.l	|- L = (LineG ` G )
tglineelsb2.g	|- ( ph -> G e. TarskiG )
tglineelsb2.1	|- ( ph -> P e. B )
tglineelsb2.2	|- ( ph -> Q e. B )
tglineelsb2.4	|- ( ph -> P =/= Q )

Assertion

Ref	Expression
tglinecom	|- ( ph -> ( P L Q ) = ( Q L P ) )

Proof of Theorem tglinecom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . . 4 |- B = ( Base ` G )
2		tglineelsb2.i	. . . 4 |- I = (Itv ` G )
3		tglineelsb2.l	. . . 4 |- L = (LineG ` G )
4		tglineelsb2.g	. . . . 5 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> G e. TarskiG )
6		tglineelsb2.2	. . . . 5 |- ( ph -> Q e. B )
7	6	adantr 481	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q e. B )
8		tglineelsb2.1	. . . . 5 |- ( ph -> P e. B )
9	8	adantr 481	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> P e. B )
10		tglineelsb2.4	. . . . . 6 |- ( ph -> P =/= Q )
11	1, 3, 2, 4, 8, 6, 10	tglnssp 25447	. . . . 5 |- ( ph -> ( P L Q ) C_ B )
12	11	sselda 3603	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. B )
13	10	necomd 2849	. . . . 5 |- ( ph -> Q =/= P )
14	13	adantr 481	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q =/= P )
15		simpr 477	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( P L Q ) )
16	1, 2, 3, 5, 7, 9, 12, 14, 15	lncom 25517	. . 3 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( Q L P ) )
17	4	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> G e. TarskiG )
18	8	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P e. B )
19	6	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> Q e. B )
20	1, 3, 2, 4, 6, 8, 13	tglnssp 25447	. . . . 5 |- ( ph -> ( Q L P ) C_ B )
21	20	sselda 3603	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. B )
22	10	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P =/= Q )
23		simpr 477	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( Q L P ) )
24	1, 2, 3, 17, 18, 19, 21, 22, 23	lncom 25517	. . 3 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( P L Q ) )
25	16, 24	impbida 877	. 2 |- ( ph -> ( x e. ( P L Q ) <-> x e. ( Q L P ) ) )
26	25	eqrdv 2620	1 |- ( ph -> ( P L Q ) = ( Q L P ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tglinethru  25531  coltr3  25543  footeq  25616  colperpexlem3  25624  mideulem2  25626  opphllem  25627  midex  25629  opphllem3  25641  opphllem5  25643  lmicom  25680  lmiisolem  25688  lnperpex  25695  trgcopy  25696