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Theorem lnssplng 28996

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 488	. . . . . 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 742	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
8		simp-4r 793	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
9		simpllr 785	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
10	2, 3, 4, 5, 7, 8, 9	elplnglnid 28987	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑎 ⊆ (𝑎𝐸𝑟))
11	1, 10	eqsstrrd 3971	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟))
12		oveq2 7404	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑋𝐿𝑌)𝐸𝑟))
13	12	eqeq2d 2773	. . . . . 6 ⊢ (𝑠 = 𝑟 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑟)))
14	9	eldifad 3916	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ 𝑃)
15	9	eldifbd 3917	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ 𝑎)
16	15, 1	neleqtrd 2884	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ¬ 𝑟 ∈ (𝑋𝐿𝑌))
17	14, 16	eldifd 3915	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
18	1	oveq1d 7411	. . . . . 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 519	. . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 = (𝑋𝐿𝑌)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
21	6	ad4antr 742	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
22	21	adantr 484	. . . . . . 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 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝐻)
25		simplr 778	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝐻 = (𝑎𝐸𝑟))
26	24, 25	eleqtrd 2864	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑎𝐸𝑟))
27	26	adantr 484	. . . . . . 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 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝐻)
30	29, 25	eleqtrd 2864	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ (𝑎𝐸𝑟))
31	30	adantr 484	. . . . . . 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 3012	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
34	33	ad5antr 744	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑌 ≠ 𝑋)
35		simp-4r 793	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑎 ∈ ran 𝐿)
36	35	adantr 484	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
37		simpllr 785	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
38	37	adantr 484	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
39		simplr 778	. . . . . . . 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 28983	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
42	2, 3, 4, 5, 6, 40, 23	plngrnssp 28983	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
43	2, 3, 4, 6, 41, 42, 32	tglinecom 28801	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
44	43	ad5antr 744	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → (𝑋𝐿𝑌) = (𝑌𝐿𝑋))
45	39, 44	neeqtrd 3026	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑋 ∈ 𝑎) → 𝑎 ≠ (𝑌𝐿𝑋))
46		simpr 488	. . . . . . 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 28995	. . . . . 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 3967	. . . . . . . 8 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ↔ (𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟)))
49	43	difeq2d 4080	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∖ (𝑋𝐿𝑌)) = (𝑃 ∖ (𝑌𝐿𝑋)))
50	43	oveq1d 7411	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑠) = ((𝑌𝐿𝑋)𝐸𝑠))
51	50	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
52	49, 51	rexeqbidv 3337	. . . . . . . 8 ⊢ (𝜑 → (∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠) ↔ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠)))
53	48, 52	anbi12d 641	. . . . . . 7 ⊢ (𝜑 → (((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)) ↔ ((𝑌𝐿𝑋) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑌𝐿𝑋))(𝑎𝐸𝑟) = ((𝑌𝐿𝑋)𝐸𝑠))))
54	53	ad5antr 744	. . . . . 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 259	. . . . 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 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝐺 ∈ TarskiG)
57	30	adantr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ∈ (𝑎𝐸𝑟))
58	26	adantr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑌 ∈ (𝑎𝐸𝑟))
59	32	ad4antr 742	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ≠ 𝑌)
60	59	adantr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑋 ≠ 𝑌)
61	35	adantr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ∈ ran 𝐿)
62	37	adantr 484	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑟 ∈ (𝑃 ∖ 𝑎))
63		simplr 778	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ 𝑎) → 𝑎 ≠ (𝑋𝐿𝑌))
64		simpr 488	. . . . . 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 28995	. . . . 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 2962	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ¬ 𝑋 = 𝑌)
67	21	adantr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝐺 ∈ TarskiG)
68	35	adantr 484	. . . . . . . 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 28985	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
70	2, 3, 4, 5, 21, 35, 37, 26	plngssp 28985	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
71	2, 3, 4, 21, 69, 70, 59	tgelrnln 28796	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
72	71	adantr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
73		simplr 778	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑎 ≠ (𝑋𝐿𝑌))
74		simprl 780	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑎)
75	69	adantr 484	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ 𝑃)
76	70	adantr 484	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ 𝑃)
77	59	adantr 484	. . . . . . . . . 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 28799	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑋𝐿𝑌))
79	74, 78	elind 4152	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 ∈ (𝑎 ∩ (𝑋𝐿𝑌)))
80		simprr 782	. . . . . . . . 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 28800	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑌 ∈ (𝑋𝐿𝑌))
82	80, 81	elind 4152	. . . . . . . 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 28809	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎)) → 𝑋 = 𝑌)
84	66, 83	mtand 825	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → ¬ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎))
85		ianor 995	. . . . . 6 ⊢ (¬ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎) ↔ (¬ 𝑋 ∈ 𝑎 ∨ ¬ 𝑌 ∈ 𝑎))
86	84, 85	sylib 220	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) ∧ 𝑎 ≠ (𝑋𝐿𝑌)) → (¬ 𝑋 ∈ 𝑎 ∨ ¬ 𝑌 ∈ 𝑎))
87	55, 65, 86	mpjaodan 971	. . . 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 3044	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
89		simpr 488	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
90	89	sseq2d 3968	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → ((𝑋𝐿𝑌) ⊆ 𝐻 ↔ (𝑋𝐿𝑌) ⊆ (𝑎𝐸𝑟)))
91	89	eqeq1d 2764	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (𝐻 = ((𝑋𝐿𝑌)𝐸𝑠) ↔ (𝑎𝐸𝑟) = ((𝑋𝐿𝑌)𝐸𝑠)))
92	91	rexbidv 3186	. . . 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 641	. . 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 259	. 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 28982	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
96	94, 95	r19.29vva 3222	1 ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∃wrex 3086 ∖ cdif 3901 ⊆ wss 3904 ran crn 5648 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 TarskiGcstrkg 28593 Itvcitv 28599 LineGclng 28600 hlGcplng 28977

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-dju 9859 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-xnn0 12555 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-hash 14344 df-word 14527 df-concat 14584 df-s1 14610 df-s2 14861 df-s3 14862 df-trkgc 28614 df-trkgb 28615 df-trkgcb 28616 df-trkgld 28618 df-trkg 28619 df-cgrg 28677 df-leg 28749 df-hlg 28767 df-mir 28823 df-rag 28864 df-perpg 28866 df-hpg 28928 df-plng 28978

This theorem is referenced by: plng3p 28997

W3C validator