Theorem elplnglnid 29065

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 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 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 2763	. . . . 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 28818	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑃)
15	3, 4, 5, 6, 8, 10, 12, 13, 14	elplng 29062	. . . 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 3943	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1102   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ∖ cdif 3902   ⊆ wss 3905   class class class wbr 5109  {copab 5173  ran crn 5662  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  TarskiGcstrkg 28696  Itvcitv 28702  LineGclng 28703  hpGchpg 29039  hlGcplng 29055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-trkg 28722  df-plng 29056

This theorem is referenced by:  lnincplng  29066  plngrotlem1  29069  lnssplnglem  29073  lnssplng  29074  dfprlng2  29197  prlngex  29201  prlngmolem2  29203  prlngmid2  29211  quadcgrprlng  29216