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Theorem lnssplng 29125

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 29116	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ⊆ (𝑎𝐸𝑟))
11	1, 10	eqsstrrd 3973	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟))
12		oveq2 7427	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑋𝐿𝑌)𝐸𝑟))
13	12	eqeq2d 2776	. . . . . 6 ⊢ (𝑠 = 𝑟 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟)))
14	9	eldifad 3918	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ 𝑃)
15	9	eldifbd 3919	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ 𝑎)
16	15, 1	neleqtrd 2887	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ (𝑋𝐿𝑌))
17	14, 16	eldifd 3917	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
18	1	oveq1d 7434	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟))
19	13, 17, 18	rspcedvdw 3586	. . . . 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 2867	. . . . . . . 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 2867	. . . . . . . 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 3015	. . . . . . . 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 29112	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
42	2, 3, 4, 5, 6, 40, 23	plngrnssp 29112	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
43	2, 3, 4, 6, 41, 42, 32	tglinecom 28959	. . . . . . . . 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 3029	. . . . . . 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 29124	. . . . . 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 3969	. . . . . . . 8 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ↔ (𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟)))
49	43	difeq2d 4081	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∖ (𝑋𝐿𝑌)) = (𝑃 ∖ (𝑌𝐿𝑋)))
50	43	oveq1d 7434	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑌𝐿𝑋)𝐸𝑠))
51	50	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝜑 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
52	49, 51	rexeqbidv 3341	. . . . . . . 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 29124	. . . . 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 2965	. . . . . . 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 29114	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
70	2, 3, 4, 5, 21, 35, 37, 26	plngssp 29114	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
71	2, 3, 4, 21, 69, 70, 59	tgelrnln 28954	. . . . . . . . 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 28957	. . . . . . . . 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 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 28958	. . . . . . . . 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 28967	. . . . . . 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 3047	. . 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 3970	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ 𝐻 ↔ (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟)))
91	89	eqeq1d 2767	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (𝐻 = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
92	91	rexbidv 3191	. . . 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 29111	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
96	94, 95	r19.29vva 3227	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 2146 ≠ wne 2960 ∃wrex 3091 ∖ cdif 3903 ⊆ wss 3906 ran crn 5664 ‘cfv 6540 (class class class)co 7419 Basecbs 17291 TarskiGcstrkg 28747 Itvcitv 28753 LineGclng 28754 hlGcplng 29106

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-n0 12520 df-xnn0 12593 df-z 12607 df-uz 12879 df-fz 13552 df-fzo 13700 df-hash 14385 df-word 14569 df-concat 14626 df-s1 14653 df-s2 14909 df-s3 14910 df-trkgc 28768 df-trkgb 28769 df-trkgcb 28770 df-trkgld 28772 df-trkg 28773 df-cgrg 28831 df-leg 28903 df-hlg 28921 df-mir 28981 df-rag 29025 df-perpg 29027 df-hpg 29091 df-plng 29107

This theorem is used by: lnssplng1 29126 plng3p 29130

W3C validator