Theorem elplnglnid 29043

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 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2	1	3mix1d 1353	. . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐺 ∈ TarskiG)
9		elplng.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
10	9	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ ran 𝐿)
11		elplng.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))
12	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑅 ∈ (𝑃 ∖ 𝐴))
13		eqid 2770	. . . . 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 28798	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑃)
15	3, 4, 5, 6, 8, 10, 12, 13, 14	elplng 29040	. . . 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 417	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴𝐸𝑅)))
18	17	ssrdv 3951	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   = wceq 1568   ∈ wcel 2150  ∃wrex 3096   ∖ cdif 3910   ⊆ wss 3913   class class class wbr 5114  {copab 5178  ran crn 5666  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  TarskiGcstrkg 28676  Itvcitv 28682  LineGclng 28683  hpGchpg 29018  hlGcplng 29033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-trkg 28702  df-plng 29034

This theorem is referenced by:  lnincplng  29044  plngrotlem1  29047  lnssplnglem  29051  lnssplng  29052  dfprlng2  29174  prlngex  29178  prlngmolem2  29180  prlngmid2  29187