Theorem tglinecom 23053

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 22998	. . . . 5 |- ( ph -> ( P L Q ) C_ B )
12	11	sselda 3368	. . . 4 |- ( ( ph /\ x e. ( P L Q ) ) -> x e. B )
13	10	necomd 2707	. . . . 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 23041	. . 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 22998	. . . . 5 |- ( ph -> ( Q L P ) C_ B )
21	20	sselda 3368	. . . 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 23041	. . 3 |- ( ( ph /\ x e. ( Q L P ) ) -> x e. ( P L Q ) )
25	16, 24	impbida 828	. 2 |- ( ph -> ( x e. ( P L Q ) <-> x e. ( Q L P ) ) )
26	25	eqrdv 2441	1 |- ( ph -> ( P L Q ) = ( Q L P ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2618   ` cfv 5430  (class class class)co 6103   Basecbs 14186  TarskiGcstrkg 22901  Itvcitv 22909  LineGclng 22910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-trkgc 22921  df-trkgb 22922  df-trkgcb 22923  df-trkg 22928

This theorem is referenced by:  tglinethru  23054