Theorem plngval 29007

Description: The plane defined by a line 𝐴 and a point 𝑅 outside of 𝐴. This is defined as the union of 3 parts: the line itself, the open half-plane containing 𝑅, and the points opposite to 𝑅 (see islnopp 28970). (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)
plngval.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
plngval.r	⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))

Assertion

Ref	Expression
plngval	⊢ (𝜑 → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})

Distinct variable groups:   𝑡,𝐴,𝑥   𝑡,𝐺,𝑥   𝑥,𝑃   𝑡,𝑅,𝑥   𝜑,𝑡,𝑥

Allowed substitution hints:   𝑃(𝑡)   𝐸(𝑥,𝑡)   𝐼(𝑥,𝑡)   𝐿(𝑥,𝑡)

Proof of Theorem plngval

Dummy variables 𝑎 𝑟 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plngval.e	. . 3 ⊢ 𝐸 = (hlG‘𝐺)
2		df-plng 29004	. . . 4 ⊢ 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))}))
3		fveq2 6871	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		plngval.1	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2818	. . . . . 6 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5919	. . . . 5 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6871	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		plngval.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	eqtr4di 2818	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10	9	difeq1d 4082	. . . . 5 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
11		biidd 265	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
12		fveq2 6871	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
13	12	fveq1d 6873	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
14	13	breqd 5116	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
15		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
16		plngval.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
17	15, 16	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
18	17	oveqd 7417	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥𝐼𝑟))
19	18	eleq2d 2851	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥𝐼𝑟)))
20	19	rexbidv 3189	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟)))
21	11, 14, 20	3orbi123d 1459	. . . . . 6 ⊢ (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))))
22	9, 21	rabeqbidv 3435	. . . . 5 ⊢ (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))})
23	6, 10, 22	mpoeq123dv 7475	. . . 4 ⊢ (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))}))
24		plngval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
25	24	elexd 3480	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
26	4	fvexi 6885	. . . . . . 7 ⊢ 𝐿 ∈ V
27	26	rnex 7895	. . . . . 6 ⊢ ran 𝐿 ∈ V
28	27	a1i 11	. . . . 5 ⊢ (𝜑 → ran 𝐿 ∈ V)
29	8	fvexi 6885	. . . . . . 7 ⊢ 𝑃 ∈ V
30	29	difexi 5291	. . . . . 6 ⊢ (𝑃 ∖ 𝑎) ∈ V
31	30	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
32	28, 31	mpoexd 8065	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}) ∈ V)
33	2, 23, 25, 32	fvmptd3 7003	. . 3 ⊢ (𝜑 → (hlG‘𝐺) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
34	1, 33	eqtrid 2812	. 2 ⊢ (𝜑 → 𝐸 = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}))
35		plngval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
36		plngval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))
37	36	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑅 ∈ (𝑃 ∖ 𝐴))
38		difeq2 4077	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑃 ∖ 𝑎) = (𝑃 ∖ 𝐴))
39	38	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑃 ∖ 𝑎) = (𝑃 ∖ 𝐴))
40	37, 39	eleqtrrd 2868	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑅 ∈ (𝑃 ∖ 𝑎))
41		eqid 2765	. . . 4 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}
42	29	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → 𝑃 ∈ V)
43	41, 42	rabexd 5301	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} ∈ V)
44		eleq2w2 2761	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝐴))
45	44	ad2antrl 740	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝐴))
46		eqidd 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → 𝑥 = 𝑥)
47		fveq2 6871	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((hpG‘𝐺)‘𝑎) = ((hpG‘𝐺)‘𝐴))
48	47	ad2antrl 740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → ((hpG‘𝐺)‘𝑎) = ((hpG‘𝐺)‘𝐴))
49		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → 𝑟 = 𝑅)
50	46, 48, 49	breq123d 5119	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → (𝑥((hpG‘𝐺)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝐴)𝑅))
51		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → 𝑎 = 𝐴)
52	49	oveq2d 7416	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → (𝑥𝐼𝑟) = (𝑥𝐼𝑅))
53	52	eleq2d 2851	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → (𝑡 ∈ (𝑥𝐼𝑟) ↔ 𝑡 ∈ (𝑥𝐼𝑅)))
54	51, 53	rexeqbidv 3340	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟) ↔ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅)))
55	45, 50, 54	3orbi123d 1459	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → ((𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))))
56	55	rabbidv 3424	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑟 = 𝑅)) → {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})
57	35, 40, 43, 56	ovmpodv2 7558	. 2 ⊢ (𝜑 → (𝐸 = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))}) → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))}))
58	34, 57	mpd 16	1 ⊢ (𝜑 → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ w3o 1100   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   class class class wbr 5105  ran crn 5653  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  TarskiGcstrkg 28654  Itvcitv 28660  LineGclng 28661  hpGchpg 28988  hlGcplng 29003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-plng 29004

This theorem is referenced by:  isplng  29008  plngrnssp  29009  elplng  29010  plngssp  29011  plngcplem  29015