Theorem elplnglnid 29254

Description: The line 𝐴 itself is a subset of a plane defined by the line 𝐴 and a point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)

Hypotheses

Ref	Expression
plngval.p	⊢ 𝑃 = (Base‘𝐺)
plngval.i	⊢ 𝐼 = (Itv‘𝐺)
plngval.1	⊢ 𝐿 = (LineG‘𝐺)
plngval.e	⊢ 𝐸 = (hlG‘𝐺)
plngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
elplng.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
elplng.r	⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))

Assertion

Ref	Expression
elplnglnid	⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

Proof of Theorem elplnglnid

Dummy variables 𝑎 𝑏 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2	1	3mix1d 1355	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∨ 𝑧((hpG‘𝐺)‘𝐴)𝑅 ∨ 𝑧{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}𝑅))
3		plngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
4		plngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5		plngval.1	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
6		plngval.e	. . . . 5 ⊢ 𝐸 = (hlG‘𝐺)
7		plngval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐺 ∈ TarskiG)
9		elplng.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ ran 𝐿)
11		elplng.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑅 ∈ (𝑃 ∖ 𝐴))
13		eqid 2761	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}
14	3, 5, 4, 8, 10, 1	tglnpt 29005	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑃)
15	3, 4, 5, 6, 8, 10, 12, 13, 14	elplng 29251	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ (𝐴𝐸𝑅) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧((hpG‘𝐺)‘𝐴)𝑅 ∨ 𝑧{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}𝑅)))
16	2, 15	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐴𝐸𝑅))
17	16	ex 418	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴𝐸𝑅)))
18	17	ssrdv 3937	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5103  {copab 5167  ran crn 5652  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  TarskiGcstrkg 28882  Itvcitv 28888  LineGclng 28889  hpGchpg 29228  hlGcplng 29244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-trkg 28908  df-plng 29245

This theorem is used by:  lnincplng  29255  plngrotlem1  29258  lnssplnglem  29262  lnssplng  29263  dfprlng2  29418  prlngex  29422  prlngmolem2  29424  prlngmid2  29432  quadcgrprlng  29437