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Theorem lnssplng 29327

Description: A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. Theorem 9.25 of [Schwabhauser] p. 75. (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)
lnssplng.h	⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
lnssplng.x	⊢ (𝜑 → 𝑋 ∈ 𝐻)
lnssplng.y	⊢ (𝜑 → 𝑌 ∈ 𝐻)
lnssplng.1	⊢ (𝜑 → 𝑋 ≠ 𝑌)

**Assertion**
Ref	Expression
lnssplng	⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))

Distinct variable groups: 𝐸,𝑠 𝐻,𝑠 𝐿,𝑠 𝑃,𝑠 𝑋,𝑠 𝑌,𝑠 𝜑,𝑠 Allowed substitution hints: 𝐺(𝑠) 𝐼(𝑠)

Proof of Theorem lnssplng Dummy variables 𝑎 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 = (𝑋𝐿𝑌))
2		plngval.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
3		plngval.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
4		plngval.1	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
5		plngval.e	. . . . . . 7 ⊢ 𝐸 = (hlG‘𝐺)
6		plngval.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
8		simp-4r 796	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
9		simpllr 788	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
10	2, 3, 4, 5, 7, 8, 9	elplnglnid 29318	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ⊆ (𝑎𝐸𝑟))
11	1, 10	eqsstrrd 3948	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟))
12		oveq2 7459	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑋𝐿𝑌)𝐸𝑟))
13	12	eqeq2d 2774	. . . . . 6 ⊢ (𝑠 = 𝑟 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟)))
14	9	eldifad 4054	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ 𝑃)
15	9	eldifbd 4055	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ 𝑎)
16	15, 1	neleqtrd 2886	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ (𝑋𝐿𝑌))
17	14, 16	eldifd 4053	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
18	1	oveq1d 7466	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟))
19	13, 17, 18	rspcedvdw 3583	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠))
20	11, 19	jca 521	. . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
21	6	ad4antr 745	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
22	21	adantr 486	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝐺 ∈ TarskiG)
23		lnssplng.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐻)
24	23	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝐻)
25		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐻 = (𝑎𝐸𝑟))
26	24, 25	eleqtrd 2866	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑎𝐸𝑟))
27	26	adantr 486	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑌 ∈ (𝑎𝐸𝑟))
28		lnssplng.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐻)
29	28	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝐻)
30	29, 25	eleqtrd 2866	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ (𝑎𝐸𝑟))
31	30	adantr 486	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑋 ∈ (𝑎𝐸𝑟))
32		lnssplng.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
33	32	necomd 3014	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
34	33	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑌 ≠ 𝑋)
35		simp-4r 796	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
36	35	adantr 486	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
37		simpllr 788	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
38	37	adantr 486	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
39		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ≠ (𝑋𝐿𝑌))
40		lnssplng.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
41	2, 3, 4, 5, 6, 40, 28	plngrnssp 29314	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
42	2, 3, 4, 5, 6, 40, 23	plngrnssp 29314	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
43	2, 3, 4, 6, 41, 42, 32	tglinecom 29160	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
44	43	ad5antr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
45	39, 44	neeqtrd 3028	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ≠ (𝑌𝐿𝑋))
46		simpr 490	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → ¬ 𝑋 ∈ 𝑎)
47	2, 3, 4, 5, 22, 27, 31, 34, 36, 38, 45, 46	lnssplnglem 29326	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → ((𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
48	43	sseq1d 3944	. . . . . . . 8 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ↔ (𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟)))
49	43	difeq2d 4066	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∖ (𝑋𝐿𝑌)) = (𝑃 ∖ (𝑌𝐿𝑋)))
50	43	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑌𝐿𝑋)𝐸𝑠))
51	50	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝜑 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
52	49, 51	rexeqbidv 3339	. . . . . . . 8 ⊢ (𝜑 → (∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
53	48, 52	anbi12d 644	. . . . . . 7 ⊢ (𝜑 → (((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)) ↔ ((𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠))))
54	53	ad5antr 747	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → (((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)) ↔ ((𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠))))
55	47, 54	mpbird 260	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
56	21	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝐺 ∈ TarskiG)
57	30	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ∈ (𝑎𝐸𝑟))
58	26	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑌 ∈ (𝑎𝐸𝑟))
59	32	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ≠ 𝑌)
60	59	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ≠ 𝑌)
61	35	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
62	37	adantr 486	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
63		simplr 781	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ≠ (𝑋𝐿𝑌))
64		simpr 490	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → ¬ 𝑌 ∈ 𝑎)
65	2, 3, 4, 5, 56, 57, 58, 60, 61, 62, 63, 64	lnssplnglem 29326	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
66	59	neneqd 2964	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ¬ 𝑋 = 𝑌)
67	21	adantr 486	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝐺 ∈ TarskiG)
68	35	adantr 486	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑎 ∈ ran 𝐿)
69	2, 3, 4, 5, 21, 35, 37, 30	plngssp 29316	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
70	2, 3, 4, 5, 21, 35, 37, 26	plngssp 29316	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
71	2, 3, 4, 21, 69, 70, 59	tgelrnln 29155	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
72	71	adantr 486	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
73		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑎 ≠ (𝑋𝐿𝑌))
74		simprl 783	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑎)
75	69	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑃)
76	70	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ 𝑃)
77	59	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ≠ 𝑌)
78	2, 3, 4, 67, 75, 76, 77	tglinerflx1 29158	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑋𝐿𝑌))
79	74, 78	elind 4100	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑎 ∩ (𝑋𝐿𝑌)))
80		simprr 785	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ 𝑎)
81	2, 3, 4, 67, 75, 76, 77	tglinerflx2 29159	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ (𝑋𝐿𝑌))
82	80, 81	elind 4100	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ (𝑎 ∩ (𝑋𝐿𝑌)))
83	2, 3, 4, 67, 68, 72, 73, 79, 82	tglineineq 29168	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 = 𝑌)
84	66, 83	mtand 828	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ¬ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎))
85		ianor 997	. . . . . 6 ⊢ (¬ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎) ↔ (¬ 𝑋 ∈ 𝑎 ∨ ¬ 𝑌 ∈ 𝑎))
86	84, 85	sylib 221	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → (¬ 𝑋 ∈ 𝑎 ∨ ¬ 𝑌 ∈ 𝑎))
87	55, 65, 86	mpjaodan 973	. . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
88	20, 87	pm2.61dane 3046	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
89		simpr 490	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
90	89	sseq2d 3945	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ 𝐻 ↔ (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟)))
91	89	eqeq1d 2765	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (𝐻 = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
92	91	rexbidv 3190	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠) ↔ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
93	90, 92	anbi12d 644	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)) ↔ ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠))))
94	88, 93	mpbird 260	. 2 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))
95	2, 3, 4, 5, 6, 40	isplng 29313	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
96	94, 95	r19.29vva 3226	1 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 ⊆ wss 3900 ∖ cdif 4048 ran crn 5659 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 TarskiGcstrkg 28946 Itvcitv 28952 LineGclng 28953 hlGcplng 29308

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-concat 14757 df-s1 14784 df-s2 15041 df-s3 15042 df-trkgc 28967 df-trkgb 28968 df-trkgcb 28969 df-trkgld 28971 df-trkg 28972 df-cgrg 29031 df-leg 29103 df-hlg 29121 df-mir 29182 df-rag 29226 df-perpg 29228 df-hpg 29293 df-plng 29309

This theorem is used by: lnssplng1 29328 plng3p 29332

W3C validator