Theorem tglinecom 28615

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tglinecom	⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG)
6		tglineelsb2.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵)
8		tglineelsb2.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵)
10		tglineelsb2.4	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
11	1, 3, 2, 4, 8, 6, 10	tglnssp 28532	. . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
12	11	sselda 3943	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵)
13	10	necomd 2980	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃)
15		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
16	1, 2, 3, 5, 7, 9, 12, 14, 15	lncom 28602	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃))
17	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG)
18	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ∈ 𝐵)
19	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄 ∈ 𝐵)
20	1, 3, 2, 4, 6, 8, 13	tglnssp 28532	. . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
21	20	sselda 3943	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵)
22	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄)
23		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
24	1, 2, 3, 17, 18, 19, 21, 22, 23	lncom 28602	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
25	16, 24	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
26	25	eqrdv 2727	1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  TarskiGcstrkg 28407  Itvcitv 28413  LineGclng 28414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-trkgc 28428  df-trkgb 28429  df-trkgcb 28430  df-trkg 28433

This theorem is referenced by:  tglinethru  28616  coltr3  28628  footeq  28704  colperpexlem3  28712  mideulem2  28714  opphllem  28715  midex  28717  opphllem3  28729  opphllem5  28731  lmicom  28768  lmiisolem  28776  lnperpex  28783  trgcopy  28784