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Theorem lnssplng 29074

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 489	. . . . . 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 744	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
8		simp-4r 795	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
9		simpllr 787	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
10	2, 3, 4, 5, 7, 8, 9	elplnglnid 29065	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ⊆ (𝑎𝐸𝑟))
11	1, 10	eqsstrrd 3972	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟))
12		oveq2 7418	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑋𝐿𝑌)𝐸𝑟))
13	12	eqeq2d 2774	. . . . . 6 ⊢ (𝑠 = 𝑟 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟)))
14	9	eldifad 3917	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ 𝑃)
15	9	eldifbd 3918	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ 𝑎)
16	15, 1	neleqtrd 2885	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ (𝑋𝐿𝑌))
17	14, 16	eldifd 3916	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
18	1	oveq1d 7425	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟))
19	13, 17, 18	rspcedvdw 3584	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠))
20	11, 19	jca 520	. . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
21	6	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
22	21	adantr 485	. . . . . . 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 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝐻)
25		simplr 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐻 = (𝑎𝐸𝑟))
26	24, 25	eleqtrd 2865	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑎𝐸𝑟))
27	26	adantr 485	. . . . . . 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 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝐻)
30	29, 25	eleqtrd 2865	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ (𝑎𝐸𝑟))
31	30	adantr 485	. . . . . . 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 3013	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
34	33	ad5antr 746	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑌 ≠ 𝑋)
35		simp-4r 795	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
36	35	adantr 485	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
37		simpllr 787	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
38	37	adantr 485	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
39		simplr 780	. . . . . . . 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 29061	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
42	2, 3, 4, 5, 6, 40, 23	plngrnssp 29061	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
43	2, 3, 4, 6, 41, 42, 32	tglinecom 28908	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
44	43	ad5antr 746	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
45	39, 44	neeqtrd 3027	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ≠ (𝑌𝐿𝑋))
46		simpr 489	. . . . . . 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 29073	. . . . . 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 3968	. . . . . . . 8 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ↔ (𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟)))
49	43	difeq2d 4081	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∖ (𝑋𝐿𝑌)) = (𝑃 ∖ (𝑌𝐿𝑋)))
50	43	oveq1d 7425	. . . . . . . . . 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 643	. . . . . . 7 ⊢ (𝜑 → (((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)) ↔ ((𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠))))
54	53	ad5antr 746	. . . . . 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 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝐺 ∈ TarskiG)
57	30	adantr 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ∈ (𝑎𝐸𝑟))
58	26	adantr 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑌 ∈ (𝑎𝐸𝑟))
59	32	ad4antr 744	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ≠ 𝑌)
60	59	adantr 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ≠ 𝑌)
61	35	adantr 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
62	37	adantr 485	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
63		simplr 780	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ≠ (𝑋𝐿𝑌))
64		simpr 489	. . . . . 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 29073	. . . . 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 2963	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ¬ 𝑋 = 𝑌)
67	21	adantr 485	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝐺 ∈ TarskiG)
68	35	adantr 485	. . . . . . . 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 29063	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
70	2, 3, 4, 5, 21, 35, 37, 26	plngssp 29063	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
71	2, 3, 4, 21, 69, 70, 59	tgelrnln 28903	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
72	71	adantr 485	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
73		simplr 780	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑎 ≠ (𝑋𝐿𝑌))
74		simprl 782	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑎)
75	69	adantr 485	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑃)
76	70	adantr 485	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ 𝑃)
77	59	adantr 485	. . . . . . . . . 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 28906	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑋𝐿𝑌))
79	74, 78	elind 4153	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑎 ∩ (𝑋𝐿𝑌)))
80		simprr 784	. . . . . . . . 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 28907	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ (𝑋𝐿𝑌))
82	80, 81	elind 4153	. . . . . . . 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 28916	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 = 𝑌)
84	66, 83	mtand 827	. . . . . 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 3045	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
89		simpr 489	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
90	89	sseq2d 3969	. . . 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 3189	. . . 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 643	. . 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 29060	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
96	94, 95	r19.29vva 3225	1 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 ∖ cdif 3902 ⊆ wss 3905 ran crn 5662 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 TarskiGcstrkg 28696 Itvcitv 28702 LineGclng 28703 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 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

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-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 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-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 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-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 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-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-fz 13531 df-fzo 13679 df-hash 14363 df-word 14547 df-concat 14604 df-s1 14630 df-s2 14881 df-s3 14882 df-trkgc 28717 df-trkgb 28718 df-trkgcb 28719 df-trkgld 28721 df-trkg 28722 df-cgrg 28780 df-leg 28852 df-hlg 28870 df-mir 28930 df-rag 28974 df-perpg 28976 df-hpg 29040 df-plng 29056

This theorem is referenced by: lnssplng1 29075 plng3p 29079

W3C validator