Theorem tglinecom 28569

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 28486	. . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
12	11	sselda 3949	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵)
13	10	necomd 2981	. . . . 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 28556	. . 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 28486	. . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
21	20	sselda 3949	. . . 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 28556	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
25	16, 24	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
26	25	eqrdv 2728	1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  TarskiGcstrkg 28361  Itvcitv 28367  LineGclng 28368

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-trkgc 28382  df-trkgb 28383  df-trkgcb 28384  df-trkg 28387

This theorem is referenced by:  tglinethru  28570  coltr3  28582  footeq  28658  colperpexlem3  28666  mideulem2  28668  opphllem  28669  midex  28671  opphllem3  28683  opphllem5  28685  lmicom  28722  lmiisolem  28730  lnperpex  28737  trgcopy  28738