Theorem elplnglnid 28987

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 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2	1	3mix1d 1350	. . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐺 ∈ TarskiG)
9		elplng.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
10	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ ran 𝐿)
11		elplng.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))
12	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑅 ∈ (𝑃 ∖ 𝐴))
13		eqid 2762	. . . . 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 28715	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑃)
15	3, 4, 5, 6, 8, 10, 12, 13, 14	elplng 28984	. . . 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 259	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐴𝐸𝑅))
17	16	ex 416	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴𝐸𝑅)))
18	17	ssrdv 3942	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1097   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ∖ cdif 3901   ⊆ wss 3904   class class class wbr 5100  {copab 5162  ran crn 5648  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  TarskiGcstrkg 28593  Itvcitv 28599  LineGclng 28600  hpGchpg 28927  hlGcplng 28977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-trkg 28619  df-plng 28978

This theorem is referenced by:  lnincplng  28988  plngrotlem1  28991  lnssplnglem  28995  lnssplng  28996