Theorem tglinecom 28707

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 28624	. . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
12	11	sselda 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵)
13	10	necomd 2987	. . . . 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 28694	. . 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 28624	. . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
21	20	sselda 3933	. . . 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 28694	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
25	16, 24	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
26	25	eqrdv 2734	1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  TarskiGcstrkg 28499  Itvcitv 28505  LineGclng 28506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-trkgc 28520  df-trkgb 28521  df-trkgcb 28522  df-trkg 28525

This theorem is referenced by:  tglinethru  28708  coltr3  28720  footeq  28796  colperpexlem3  28804  mideulem2  28806  opphllem  28807  midex  28809  opphllem3  28821  opphllem5  28823  lmicom  28860  lmiisolem  28868  lnperpex  28875  trgcopy  28876