Theorem elplnglnid 29270

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 2760	. . . . 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 29021	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑃)
15	3, 4, 5, 6, 8, 10, 12, 13, 14	elplng 29267	. . . 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 3915	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3896   ∖ cdif 4044   class class class wbr 5102  {copab 5166  ran crn 5652  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  TarskiGcstrkg 28898  Itvcitv 28904  LineGclng 28905  hpGchpg 29244  hlGcplng 29260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  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 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-trkg 28924  df-plng 29261

This theorem is used by:  lnincplng  29271  plngrotlem1  29274  lnssplnglem  29278  lnssplng  29279  dfprlng2  29434  prlngex  29438  prlngmolem2  29440  prlngmid2  29448  quadcgrprlng  29453