Theorem isplng 29265

Description: The property of being a plane. (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)
isplng.h	⊢ (𝜑 → 𝐻 ∈ ran 𝐸)

Assertion

Ref	Expression
isplng	⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))

Distinct variable groups:   𝐺,𝑎,𝑟   𝐻,𝑎,𝑟   𝐿,𝑎,𝑟   𝑃,𝑟   𝜑,𝑎,𝑟

Allowed substitution hints:   𝑃(𝑎)   𝐸(𝑟, 𝑎)   𝐼(𝑟, 𝑎)

Proof of Theorem isplng

Dummy variables 𝑔 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isplng.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
2		plngval.e	. . . . . 6 ⊢ 𝐸 = (hlG‘𝐺)
3		df-plng 29261	. . . . . . 7 ⊢ hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))
4		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
5		plngval.1	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
6	4, 5	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
7	6	rneqd 5921	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
8		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
9		plngval.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
10	8, 9	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
11	10	difeq1d 4061	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
12		biidd 265	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
13		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
14	13	fveq1d 6892	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
15	14	breqd 5113	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
16		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
17		plngval.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (Itv‘𝐺)
18	16, 17	eqtr4di 2813	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
19	18	oveqd 7441	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥𝐼𝑟))
20	19	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥𝐼𝑟)))
21	20	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟)))
22	12, 15, 21	3orbi123d 1463	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟)) ↔ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))))
23	10, 22	rabeqbidv 3429	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))} = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
24	7, 11, 23	mpoeq123dv 7499	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
25		plngval.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
26	25	elexd 3473	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ V)
27	5	fvexi 6904	. . . . . . . . . 10 ⊢ 𝐿 ∈ V
28	27	rnex 7926	. . . . . . . . 9 ⊢ ran 𝐿 ∈ V
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ran 𝐿 ∈ V)
30	9	fvexi 6904	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
31	30	difexi 5292	. . . . . . . . 9 ⊢ (𝑃 ∖ 𝑎) ∈ V
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
33	29, 32	mpoexd 8097	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}) ∈ V)
34	3, 24, 26, 33	fvmptd3 7022	. . . . . 6 ⊢ (𝜑 → (hlG‘𝐺) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
35	2, 34	eqtrid 2807	. . . . 5 ⊢ (𝜑 → 𝐸 = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
36	35	rneqd 5921	. . . 4 ⊢ (𝜑 → ran 𝐸 = ran (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
37	1, 36	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝐻 ∈ ran (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
38		eqid 2760	. . . 4 ⊢ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
39	30	rabex 5300	. . . 4 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} ∈ V
40	38, 39	elrnmpo 7560	. . 3 ⊢ (𝐻 ∈ ran (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}) ↔ ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
41	37, 40	sylib 221	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
42	25	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → 𝐺 ∈ TarskiG)
43		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → 𝑎 ∈ ran 𝐿)
44		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
45	9, 17, 5, 2, 42, 43, 44	plngval 29264	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → (𝑎𝐸𝑟) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
46	45	eqeq2d 2771	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → (𝐻 = (𝑎𝐸𝑟) ↔ 𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
47	46	biimprd 251	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) → (𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} → 𝐻 = (𝑎𝐸𝑟)))
48	47	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ran 𝐿 ∧ 𝑟 ∈ (𝑃 ∖ 𝑎))) → (𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} → 𝐻 = (𝑎𝐸𝑟)))
49	48	reximdvva 3210	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟)))
50	41, 49	mpd 16	1 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450   ∖ cdif 4044   class class class wbr 5102  ran crn 5652  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  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-plng 29261

This theorem is used by:  plngrnssp  29266  lnssplng  29279