Theorem tglinecom 24680

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 467	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> G e. TarskiG )
6		tglineelsb2.2	. . . . 5 |- ( ph -> Q e. B )
7	6	adantr 467	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q e. B )
8		tglineelsb2.1	. . . . 5 |- ( ph -> P e. B )
9	8	adantr 467	. . . 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 24597	. . . . 5 |- ( ph -> ( P L Q ) C_ B )
12	11	sselda 3432	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. B )
13	10	necomd 2679	. . . . 5 |- ( ph -> Q =/= P )
14	13	adantr 467	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q =/= P )
15		simpr 463	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( P L Q ) )
16	1, 2, 3, 5, 7, 9, 12, 14, 15	lncom 24667	. . 3 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( Q L P ) )
17	4	adantr 467	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> G e. TarskiG )
18	8	adantr 467	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P e. B )
19	6	adantr 467	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> Q e. B )
20	1, 3, 2, 4, 6, 8, 13	tglnssp 24597	. . . . 5 |- ( ph -> ( Q L P ) C_ B )
21	20	sselda 3432	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. B )
22	10	adantr 467	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P =/= Q )
23		simpr 463	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( Q L P ) )
24	1, 2, 3, 17, 18, 19, 21, 22, 23	lncom 24667	. . 3 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( P L Q ) )
25	16, 24	impbida 843	. 2 |- ( ph -> ( x e. ( P L Q ) <-> x e. ( Q L P ) ) )
26	25	eqrdv 2449	1 |- ( ph -> ( P L Q ) = ( Q L P ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   ` cfv 5582  (class class class)co 6290   Basecbs 15121  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501

This theorem is referenced by:  tglinethru  24681  coltr3  24693  footeq  24766  colperpexlem3  24774  mideulem2  24776  opphllem  24777  midex  24779  opphllem3  24791  opphllem5  24793  lmicom  24830  lmiisolem  24838  lnperpex  24845  trgcopy  24846