Theorem quadcgrprlng 29194

Description: Nontrivial quadrilaterals with congruent and parallel opposite sides are parallelograms. Theorem 12.20 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)

Hypotheses

Ref	Expression
quadcgrprlng.p	⊢ 𝑃 = (Base‘𝐺)
quadcgrprlng.d	⊢ − = (dist‘𝐺)
quadcgrprlng.i	⊢ 𝐼 = (Itv‘𝐺)
quadcgrprlng.l	⊢ 𝐿 = (LineG‘𝐺)
quadcgrprlng.r	⊢ ∥ = (parlnG‘𝐺)
quadcgrprlng.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
quadcgrprlng.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
quadcgrprlng.1	⊢ (𝜑 → 𝐺 ∈ TarskiGE)
quadcgrprlng.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
quadcgrprlng.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
quadcgrprlng.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
quadcgrprlng.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
quadcgrprlng.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
quadcgrprlng.3	⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
quadcgrprlng.4	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
quadcgrprlng.5	⊢ (𝜑 → 𝑌𝑂𝑊)

Assertion

Ref	Expression
quadcgrprlng	⊢ (𝜑 → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))

Distinct variable groups:   − ,𝑎,𝑡   ∥ ,𝑎,𝑡   𝐺,𝑎,𝑏,𝑡   𝐼,𝑎,𝑡,𝑏   𝐿,𝑎,𝑡,𝑏   𝑂,𝑎,𝑡,𝑏   𝑃,𝑎,𝑡,𝑏   𝑊,𝑎,𝑡   𝑋,𝑎,𝑡,𝑏   𝑌,𝑎,𝑡,𝑏   𝑍,𝑎,𝑡,𝑏   𝜑,𝑎,𝑡

Allowed substitution hints:   𝜑(𝑏)   ∥ (𝑏)   − (𝑏)   𝑊(𝑏)

Proof of Theorem quadcgrprlng

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		quadcgrprlng.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
2		eqid 2763	. . . . 5 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
3		quadcgrprlng.r	. . . . 5 ⊢ ∥ = (parlnG‘𝐺)
4		quadcgrprlng.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝐺 ∈ TarskiG)
6		quadcgrprlng.1	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiGE)
7	6	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝐺 ∈ TarskiGE)
8		quadcgrprlng.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
9		quadcgrprlng.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
10		quadcgrprlng.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
11		quadcgrprlng.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
12		quadcgrprlng.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
13		quadcgrprlng.2	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
14	8, 1, 9, 4, 10, 11, 12, 13	ncolrot2 28810	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	8, 9, 1, 4, 11, 12, 10, 14	ncolne2 28877	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
16	15	necomd 3013	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
17	8, 9, 1, 4, 10, 11, 16	tgelrnln 28881	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
18	17	ad3antrrr 742	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑌𝐿𝑍) ∈ ran 𝐿)
19	13	orsild 1019	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
20	12, 19	eldifd 3917	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))
21	20	ad3antrrr 742	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝑋 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))
22	8, 1, 2, 5, 18, 21	tgelrnpln 29036	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋) ∈ ran (hlG‘𝐺))
23		quadcgrprlng.3	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
24	1, 3, 4, 23	prlngrcl2 29171	. . . . . 6 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
25	24	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑍𝐿𝑊) ∈ ran 𝐿)
26	8, 9, 1, 4, 10, 11, 16	tglinerflx2 28885	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑍))
27		quadcgrprlng.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
28	8, 9, 1, 4, 11, 27, 24	tglnne 28879	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
29	8, 9, 1, 4, 11, 27, 28	tglinerflx1 28884	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
30	26, 29	elind 4154	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ((𝑌𝐿𝑍) ∩ (𝑍𝐿𝑊)))
31	30	ne0d 4296	. . . . . 6 ⊢ (𝜑 → ((𝑌𝐿𝑍) ∩ (𝑍𝐿𝑊)) ≠ ∅)
32	31	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → ((𝑌𝐿𝑍) ∩ (𝑍𝐿𝑊)) ≠ ∅)
33	14	orsild 1019	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
34	26	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝑍 ∈ (𝑌𝐿𝑍))
35	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiG)
36	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiGE)
37	1, 2, 3, 4, 23	prlngsym 29169	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍𝐿𝑊) ∥ (𝑋𝐿𝑌))
38	37	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑍𝐿𝑊) ∥ (𝑋𝐿𝑌))
39	24	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑍𝐿𝑊) ∈ ran 𝐿)
40	1, 2, 3, 35, 39	prlngref 29168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑍𝐿𝑊) ∥ (𝑍𝐿𝑊))
41		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑊))
42	40, 41	breqtrrd 5140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑍𝐿𝑊) ∥ (𝑌𝐿𝑍))
43	8, 9, 1, 4, 12, 10, 11, 13	ncolne1 28876	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
44	8, 9, 1, 4, 12, 10, 43	tglinerflx2 28885	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐿𝑌))
45	44	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝑌 ∈ (𝑋𝐿𝑌))
46	8, 9, 1, 4, 10, 11, 16	tglinerflx1 28884	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑍))
47	46	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝑌 ∈ (𝑌𝐿𝑍))
48	8, 3, 35, 36, 38, 42, 45, 47	prlngeq 29185	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) = (𝑌𝐿𝑍))
49	34, 48	eleqtrrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑌𝐿𝑍) = (𝑍𝐿𝑊)) → 𝑍 ∈ (𝑋𝐿𝑌))
50	33, 49	mtand 827	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑌𝐿𝑍) = (𝑍𝐿𝑊))
51	50	neqned 2965	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) ≠ (𝑍𝐿𝑊))
52	51	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑌𝐿𝑍) ≠ (𝑍𝐿𝑊))
53		simplr 780	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑌𝐿𝑍) ∥ 𝑎)
54	8, 9, 1, 2, 5, 18, 21	elplnglnid 29043	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑌𝐿𝑍) ⊆ ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋))
55		simpr 489	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝑋 ∈ 𝑎)
56	19	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
57		nelne1 3055	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑎 ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑍)) → 𝑎 ≠ (𝑌𝐿𝑍))
58	55, 56, 57	syl2anc 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝑎 ≠ (𝑌𝐿𝑍))
59	58	necomd 3013	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑌𝐿𝑍) ≠ 𝑎)
60	1, 2, 3, 5, 53, 59, 55	prlngpln3 29177	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → 𝑎 ⊆ ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋))
61	8, 9, 1, 4, 12, 10, 11, 27, 13	tglineneq 28896	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
62	1, 2, 3, 4, 23, 61, 29	prlngpln3 29177	. . . . . . 7 ⊢ (𝜑 → (𝑍𝐿𝑊) ⊆ ((𝑋𝐿𝑌)(hlG‘𝐺)𝑍))
63	8, 1, 9, 4, 10, 11, 12, 13	ncolcom 28808	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
64	63	orsild 1019	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑍𝐿𝑌))
65	12, 64	eldifd 3917	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
66	11, 33	eldifd 3917	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
67	8, 9, 1, 2, 4, 65, 10, 66, 43	plngrot 29050	. . . . . . . 8 ⊢ (𝜑 → ((𝑋𝐿𝑌)(hlG‘𝐺)𝑍) = ((𝑍𝐿𝑌)(hlG‘𝐺)𝑋))
68	8, 9, 1, 4, 11, 10, 15	tglinecom 28886	. . . . . . . . 9 ⊢ (𝜑 → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
69	68	oveq1d 7427	. . . . . . . 8 ⊢ (𝜑 → ((𝑍𝐿𝑌)(hlG‘𝐺)𝑋) = ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋))
70	67, 69	eqtr2d 2799	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋) = ((𝑋𝐿𝑌)(hlG‘𝐺)𝑍))
71	62, 70	sseqtrrd 3975	. . . . . 6 ⊢ (𝜑 → (𝑍𝐿𝑊) ⊆ ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋))
72	71	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑍𝐿𝑊) ⊆ ((𝑌𝐿𝑍)(hlG‘𝐺)𝑋))
73	1, 2, 3, 5, 7, 22, 25, 32, 52, 53, 54, 60, 72	prlnginn0 29188	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → (𝑎 ∩ (𝑍𝐿𝑊)) ≠ ∅)
74		simpllr 787	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑌𝐿𝑍) ∥ 𝑎)
75	5	adantr 485	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝐺 ∈ TarskiG)
76	25	adantr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑍𝐿𝑊) ∈ ran 𝐿)
77		simpr 489	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊)))
78	77	elin2d 4159	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ∈ (𝑍𝐿𝑊))
79	8, 1, 9, 75, 76, 78	tglnpt 28796	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ∈ 𝑃)
80	12	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑋 ∈ 𝑃)
81	29	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝑍 ∈ (𝑍𝐿𝑊))
82	4	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiG)
83	6	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiGE)
84	8, 9, 1, 4, 12, 10, 43	tgelrnln 28881	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
85	84	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
86	1, 2, 3, 82, 85	prlngref 29168	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∥ (𝑋𝐿𝑌))
87	23	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
88	8, 9, 1, 4, 12, 10, 43	tglinerflx1 28884	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑌))
89	88	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑌))
90		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑍𝐿𝑊))
91	8, 3, 82, 83, 86, 87, 89, 90	prlngeq 29185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
92	81, 91	eleqtrrd 2866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝑍 ∈ (𝑋𝐿𝑌))
93	33, 92	mtand 827	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑍𝐿𝑊))
94	93	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ¬ 𝑋 ∈ (𝑍𝐿𝑊))
95		nelne2 3056	. . . . . . . . 9 ⊢ ((𝑤 ∈ (𝑍𝐿𝑊) ∧ ¬ 𝑋 ∈ (𝑍𝐿𝑊)) → 𝑤 ≠ 𝑋)
96	78, 94, 95	syl2anc 595	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ≠ 𝑋)
97		simp-4r 795	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑎 ∈ ran 𝐿)
98	77	elin1d 4158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ∈ 𝑎)
99		simplr 780	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑋 ∈ 𝑎)
100	8, 9, 1, 75, 79, 80, 96, 96, 97, 98, 99	tglinethru 28887	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑎 = (𝑤𝐿𝑋))
101	74, 100	breqtrd 5138	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑌𝐿𝑍) ∥ (𝑤𝐿𝑋))
102		quadcgrprlng.d	. . . . . . . 8 ⊢ − = (dist‘𝐺)
103		eqid 2763	. . . . . . . 8 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
104	11	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑍 ∈ 𝑃)
105	10	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑌 ∈ 𝑃)
106	27	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑊 ∈ 𝑃)
107	28	necomd 3013	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ≠ 𝑍)
108	107	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑊 ≠ 𝑍)
109	43	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑋 ≠ 𝑌)
110		quadcgrprlng.o	. . . . . . . . 9 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
111	8, 9, 1, 4, 12, 10, 11, 13	ncolne2 28877	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
112	8, 9, 1, 4, 12, 11, 111	tgelrnln 28881	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
113	112	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑋𝐿𝑍) ∈ ran 𝐿)
114		quadcgrprlng.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌𝑂𝑊)
115	8, 102, 9, 110, 1, 112, 4, 10, 27, 114	oppcom 29003	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊𝑂𝑌)
116	115	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑊𝑂𝑌)
117	8, 9, 1, 4, 12, 11, 111	tglinerflx2 28885	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑍))
118	117	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑍 ∈ (𝑋𝐿𝑍))
119	7	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝐺 ∈ TarskiGE)
120	13	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
121	23	ad4antr 744	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
122	19	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
123		simpllr 787	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑋 ∈ 𝑎)
124	75	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝐺 ∈ TarskiG)
125	119	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝐺 ∈ TarskiGE)
126	1, 2, 3, 4, 17	prlngref 29168	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑌𝐿𝑍))
127	126	ad5antr 746	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → (𝑌𝐿𝑍) ∥ (𝑌𝐿𝑍))
128		simp-4r 795	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → (𝑌𝐿𝑍) ∥ 𝑎)
129	26	ad5antr 746	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑍 ∈ (𝑌𝐿𝑍))
130		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑍 = 𝑤)
131	98	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑤 ∈ 𝑎)
132	130, 131	eqeltrd 2863	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑍 ∈ 𝑎)
133	8, 3, 124, 125, 127, 128, 129, 132	prlngeq 29185	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → (𝑌𝐿𝑍) = 𝑎)
134	123, 133	eleqtrrd 2866	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) ∧ 𝑍 = 𝑤) → 𝑋 ∈ (𝑌𝐿𝑍))
135	122, 134	mtand 827	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ¬ 𝑍 = 𝑤)
136	135	neqned 2965	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑍 ≠ 𝑤)
137	29	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑍 ∈ (𝑍𝐿𝑊))
138	8, 9, 1, 75, 104, 79, 136, 136, 76, 137, 78	tglinethru 28887	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑍𝐿𝑊) = (𝑍𝐿𝑤))
139	121, 138	breqtrd 5138	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑤))
140	8, 102, 1, 3, 75, 119, 80, 105, 104, 79, 120, 139, 101, 110, 9	prlngsymquadopp 29193	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤𝑂𝑌)
141	8, 9, 1, 4, 11, 27, 28	tglinecom 28886	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍𝐿𝑊) = (𝑊𝐿𝑍))
142	141	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑍𝐿𝑊) = (𝑊𝐿𝑍))
143	78, 142	eleqtrd 2865	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 ∈ (𝑊𝐿𝑍))
144	8, 9, 1, 110, 103, 75, 113, 106, 105, 116, 118, 140, 143	hlopp 29032	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤((hlG‘𝐺)‘𝑍)𝑊)
145	8, 9, 103, 27, 12, 11, 4, 107	hlid 28859	. . . . . . . . 9 ⊢ (𝜑 → 𝑊((hlG‘𝐺)‘𝑍)𝑊)
146	145	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑊((hlG‘𝐺)‘𝑍)𝑊)
147	8, 102, 1, 3, 75, 119, 80, 105, 104, 79, 120, 139, 101	prlngsymquad 29192	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ((𝑋 − 𝑌) = (𝑍 − 𝑤) ∧ (𝑌 − 𝑍) = (𝑤 − 𝑋)))
148	147	simpld 499	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑋 − 𝑌) = (𝑍 − 𝑤))
149	148	eqcomd 2769	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑍 − 𝑤) = (𝑋 − 𝑌))
150		quadcgrprlng.4	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
151	150	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 − 𝑊) = (𝑋 − 𝑌))
152	151	ad4antr 744	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑍 − 𝑊) = (𝑋 − 𝑌))
153	8, 102, 103, 104, 80, 105, 75, 106, 108, 109, 144, 146, 149, 152	hlcgreq 28869	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → 𝑤 = 𝑊)
154	153	oveq1d 7427	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑤𝐿𝑋) = (𝑊𝐿𝑋))
155	101, 154	breqtrd 5138	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
156	147	simprd 500	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑌 − 𝑍) = (𝑤 − 𝑋))
157	153	oveq1d 7427	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑤 − 𝑋) = (𝑊 − 𝑋))
158	156, 157	eqtrd 2798	. . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → (𝑌 − 𝑍) = (𝑊 − 𝑋))
159	155, 158	jca 520	. . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) ∧ 𝑤 ∈ (𝑎 ∩ (𝑍𝐿𝑊))) → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
160	73, 159	n0limd 4309	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑌𝐿𝑍) ∥ 𝑎) ∧ 𝑋 ∈ 𝑎) → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
161	160	anasss 471	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ ((𝑌𝐿𝑍) ∥ 𝑎 ∧ 𝑋 ∈ 𝑎)) → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
162	8, 1, 3, 4, 17, 12	prlngex 29179	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿((𝑌𝐿𝑍) ∥ 𝑎 ∧ 𝑋 ∈ 𝑎))
163	161, 162	r19.29a 3173	1 ⊢ (𝜑 → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287   class class class wbr 5110  {copab 5174  ran crn 5664  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  distcds 17320  TarskiGcstrkg 28674  TarskiG_Ecstrkge 28679  Itvcitv 28680  LineGclng 28681  hlGchlg 28847  hlGcplng 29033  parlnGcprlng 29164

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 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

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 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-trkgc 28695  df-trkgb 28696  df-trkgcb 28697  df-trkge 28698  df-trkgld 28699  df-trkg 28700  df-cgrg 28758  df-ismt 28780  df-leg 28830  df-hlg 28848  df-mir 28908  df-rag 28952  df-perpg 28954  df-hpg 29018  df-plng 29034  df-mid 29061  df-lmi 29062  df-cgra 29097  df-prlng 29165

This theorem is referenced by:  tgaltai  29195