Theorem tglinecom 24141

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 465	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> G e. TarskiG )
6		tglineelsb2.2	. . . . 5 |- ( ph -> Q e. B )
7	6	adantr 465	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q e. B )
8		tglineelsb2.1	. . . . 5 |- ( ph -> P e. B )
9	8	adantr 465	. . . 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 24065	. . . . 5 |- ( ph -> ( P L Q ) C_ B )
12	11	sselda 3499	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. B )
13	10	necomd 2728	. . . . 5 |- ( ph -> Q =/= P )
14	13	adantr 465	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> Q =/= P )
15		simpr 461	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( P L Q ) )
16	1, 2, 3, 5, 7, 9, 12, 14, 15	lncom 24128	. . 3 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. ( Q L P ) )
17	4	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> G e. TarskiG )
18	8	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P e. B )
19	6	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> Q e. B )
20	1, 3, 2, 4, 6, 8, 13	tglnssp 24065	. . . . 5 |- ( ph -> ( Q L P ) C_ B )
21	20	sselda 3499	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. B )
22	10	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> P =/= Q )
23		simpr 461	. . . 4 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( Q L P ) )
24	1, 2, 3, 17, 18, 19, 21, 22, 23	lncom 24128	. . 3 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( P L Q ) )
25	16, 24	impbida 832	. 2 |- ( ph -> ( x e. ( P L Q ) <-> x e. ( Q L P ) ) )
26	25	eqrdv 2454	1 |- ( ph -> ( P L Q ) = ( Q L P ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   ` cfv 5594  (class class class)co 6296   Basecbs 14644  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976

This theorem is referenced by:  tglinethru  24142  coltr3  24155  footeq  24224  colperpexlem3  24232  mideulem2  24234  opphllem  24235  midex  24237  opphllem3  24247  opphllem5  24249  lmicom  24280  lmiisolem  24287