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Theorem plngrotlem2 29323

Description: Lemma for plngrot 29325. (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)
plngrot.x	⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
plngrot.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
plngrot.z	⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
plngrot.1	⊢ (𝜑 → 𝑋 ≠ 𝑌)
plngrotlem2.4	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}
plngrotlem2.1	⊢ (𝜑 → 𝑊 ∈ 𝑃)
plngrotlem2.2	⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))
plngrotlem2.3	⊢ (𝜑 → 𝑌 ≠ 𝑊)

**Assertion**
Ref	Expression
plngrotlem2	⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))

Distinct variable groups: 𝑡,𝐸 𝐺,𝑎,𝑏,𝑡 𝐼,𝑎,𝑏,𝑡 𝐿,𝑎,𝑏,𝑡 𝑂,𝑎,𝑏,𝑡 𝑃,𝑎,𝑏,𝑡 𝑊,𝑎,𝑏,𝑡 𝑋,𝑎,𝑏,𝑡 𝑌,𝑎,𝑏,𝑡 𝑍,𝑎,𝑏,𝑡 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑎, 𝑏) 𝐸(𝑎, 𝑏)

Proof of Theorem plngrotlem2 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		plngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
3		plngval.1	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		plngval.e	. . . . 5 ⊢ 𝐸 = (hlG‘𝐺)
5		plngval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝐺 ∈ TarskiG)
7		plngrot.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
8	7	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
9		plngrot.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑌 ∈ 𝑃)
11		plngrot.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
12	11	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
13		plngrot.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
14	13	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑋 ≠ 𝑌)
15		plngrotlem2.4	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}
16		plngrotlem2.1	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
17	16	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑊 ∈ 𝑃)
18		plngrotlem2.2	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))
19	18	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑌 ∈ (𝑍𝐼𝑊))
20		plngrotlem2.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
21	20	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑌 ≠ 𝑊)
22		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍))
23	22	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍))
24		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍))
25	1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 17, 19, 21, 23, 24	plngrotlem1 29322	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
26	5	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝐺 ∈ TarskiG)
27	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
28	11	eldifad 4054	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
29	28	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ∈ 𝑃)
30	9	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ∈ 𝑃)
31	7	eldifad 4054	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
32	1, 2, 3, 5, 31, 9, 13	tglinerflx2 29159	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐿𝑌))
33		elndif 4072	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (𝑋𝐿𝑌) → ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
34	32, 33	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
35		nelne2 3057	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) → 𝑍 ≠ 𝑌)
36	11, 34, 35	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
37	36	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ≠ 𝑌)
38	1, 2, 3, 26, 29, 30, 37	tglinecom 29160	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
39	16	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑊 ∈ 𝑃)
40	20	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ≠ 𝑊)
41	1, 2, 3, 5, 9, 16, 28, 20, 18	btwnlng2 29145	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑊))
42	41	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ∈ (𝑌𝐿𝑊))
43	1, 2, 3, 26, 30, 39, 40, 29, 37, 42	tglineelsb2 29157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑌𝐿𝑊) = (𝑌𝐿𝑍))
44	1, 2, 3, 26, 30, 39, 40	tglinecom 29160	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑌𝐿𝑊) = (𝑊𝐿𝑌))
45	38, 43, 44	3eqtr2d 2804	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑍𝐿𝑌) = (𝑊𝐿𝑌))
46	45	difeq2d 4066	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑃 ∖ (𝑍𝐿𝑌)) = (𝑃 ∖ (𝑊𝐿𝑌)))
47	27, 46	eleqtrd 2866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ∈ (𝑃 ∖ (𝑊𝐿𝑌)))
48	20	neneqd 2964	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑌 = 𝑊)
49	5	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
50	31	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
51	9	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
52	13	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑋 ≠ 𝑌)
53	1, 2, 3, 49, 50, 51, 52	tgelrnln 29155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
54	28	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ∈ 𝑃)
55	36	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ≠ 𝑌)
56	1, 2, 3, 49, 54, 51, 55	tgelrnln 29155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ∈ ran 𝐿)
57	1, 2, 3, 5, 28, 9, 36	tglinerflx1 29158	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑌))
58	57	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ∈ (𝑍𝐿𝑌))
59	11	eldifbd 4055	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
60	59	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
61		nelne1 3056	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ (𝑍𝐿𝑌) ∧ ¬ 𝑍 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ≠ (𝑋𝐿𝑌))
62	58, 60, 61	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ≠ (𝑋𝐿𝑌))
63	62	necomd 3014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑌))
64	32	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑋𝐿𝑌))
65	1, 2, 3, 49, 54, 51, 55	tglinerflx2 29159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑍𝐿𝑌))
66	64, 65	elind 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑌)))
67		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ (𝑋𝐿𝑌))
68	16	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ 𝑃)
69	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑍𝐼𝑊))
70	1, 2, 3, 49, 54, 51, 68, 55, 69	btwnlng3 29146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ (𝑍𝐿𝑌))
71	67, 70	elind 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑌)))
72	1, 2, 3, 49, 53, 56, 63, 66, 71	tglineineq 29168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 = 𝑊)
73	48, 72	mtand 828	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
74	73	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
75	39, 74	eldifd 4053	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑊 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
76	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ≠ 𝑌)
77		eqid 2763	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
78	1, 77, 2, 5, 28, 9, 16, 18	tgbtwncom 29008	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑊𝐼𝑍))
79	78	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ∈ (𝑊𝐼𝑍))
80	37	necomd 3014	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ≠ 𝑍)
81	1, 77, 2, 15, 16, 28, 32, 73, 59, 78	islnoppd 29273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊𝑂𝑍)
82	81	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑊𝑂𝑍)
83	5	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝐺 ∈ TarskiG)
84	83	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝐺 ∈ TarskiG)
85	31	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑋 ∈ 𝑃)
86	9	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑌 ∈ 𝑃)
87	13	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑋 ≠ 𝑌)
88	1, 2, 3, 83, 85, 86, 87	tgelrnln 29155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
89	88	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑋𝐿𝑌) ∈ ran 𝐿)
90	11	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
91	1, 2, 3, 4, 83, 88, 90, 22	plngssp 29316	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ 𝑃)
92	91	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠 ∈ 𝑃)
93	16	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑊 ∈ 𝑃)
94	28	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍 ∈ 𝑃)
95		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠𝑂𝑍)
96	1, 2, 3, 15, 84, 89, 92, 93, 94, 95	lnopp2hpgb 29298	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑍 ↔ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
97	82, 96	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
98	97	orcd 887	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))
99	98	ex 418	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → (𝑠𝑂𝑍 → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
100	99	ex 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠𝑂𝑍 → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
101	100	a2d 30	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
102		df-or 862	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍))
103		df-or 862	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
104	101, 102, 103	3imtr4g 299	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
105	104	imp 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
106		3orass 1106	. . . . . . . 8 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
107	105, 106	sylibr 237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))
108	88	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
109	91	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ 𝑃)
110	1, 2, 3, 4, 26, 108, 75, 15, 109	elplng 29315	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑊) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
111	107, 110	mpbird 260	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑊))
112	32	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑌 ∈ (𝑋𝐿𝑌))
113	59	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
114	73	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
115	18	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑌 ∈ (𝑍𝐼𝑊))
116	1, 77, 2, 15, 94, 93, 112, 113, 114, 115	islnoppd 29273	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍𝑂𝑊)
117	1, 2, 3, 15, 84, 89, 94, 93, 116	lnoppnhpg 29299	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
118	89	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → (𝑋𝐿𝑌) ∈ ran 𝐿)
119	84	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝐺 ∈ TarskiG)
120	93	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑊 ∈ 𝑃)
121	92	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑠 ∈ 𝑃)
122		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑊𝑂𝑠)
123	1, 77, 2, 15, 3, 118, 119, 120, 121, 122	oppcom 29277	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑠𝑂𝑊)
124	89	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → (𝑋𝐿𝑌) ∈ ran 𝐿)
125	84	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝐺 ∈ TarskiG)
126	92	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑠 ∈ 𝑃)
127	93	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑊 ∈ 𝑃)
128		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑠𝑂𝑊)
129	1, 77, 2, 15, 3, 124, 125, 126, 127, 128	oppcom 29277	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑊𝑂𝑠)
130	123, 129	impbida 813	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑠 ↔ 𝑠𝑂𝑊))
131	1, 77, 2, 15, 3, 89, 84, 92, 94, 95	oppcom 29277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍𝑂𝑠)
132	1, 2, 3, 15, 84, 89, 94, 93, 92, 131	lnopp2hpgb 29298	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑠 ↔ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
133	130, 132	bitr3d 284	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑠𝑂𝑊 ↔ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
134	117, 133	mtbird 328	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑠𝑂𝑊)
135	134	ex 418	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → (𝑠𝑂𝑍 → ¬ 𝑠𝑂𝑊))
136	135	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠𝑂𝑍 → ¬ 𝑠𝑂𝑊)))
137	136	a2d 30	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊)))
138	137	imp 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊))
139	102, 138	sylan2b 606	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊))
140	139	imp 412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → ¬ 𝑠𝑂𝑊)
141		df-3or 1104	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ∨ 𝑠𝑂𝑊))
142		orcom 884	. . . . . . . . . . . . . . 15 ⊢ (((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ∨ 𝑠𝑂𝑊) ↔ (𝑠𝑂𝑊 ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
143		df-or 862	. . . . . . . . . . . . . . 15 ⊢ ((𝑠𝑂𝑊 ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)) ↔ (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
144	141, 142, 143	3bitri 300	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
145	107, 144	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
146	145	imp 412	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
147		df-or 862	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
148	146, 147	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
149	148	imp 412	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
150	149	an32s 665	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ ¬ 𝑠𝑂𝑊) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
151	140, 150	mpdan 700	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
152	151	ex 418	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
153	152, 147	sylibr 237	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
154	1, 2, 3, 4, 26, 47, 30, 75, 76, 15, 29, 79, 80, 111, 153	plngrotlem1 29322	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑊𝐿𝑌)𝐸𝑋))
155	45	eqcomd 2769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑊𝐿𝑌) = (𝑍𝐿𝑌))
156	155	oveq1d 7466	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → ((𝑊𝐿𝑌)𝐸𝑋) = ((𝑍𝐿𝑌)𝐸𝑋))
157	154, 156	eleqtrd 2866	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
158	1, 2, 3, 4, 83, 88, 90, 15, 91	elplng 29315	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)))
159	22, 158	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍))
160		3orass 1106	. . . . . 6 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)))
161		orordi 942	. . . . . 6 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
162	160, 161	bitri 278	. . . . 5 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
163	159, 162	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
164	25, 157, 163	mpjaodan 973	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
165	164	ex 418	. 2 ⊢ (𝜑 → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋)))
166	165	ssrdv 3919	1 ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 ⊆ wss 3900 ∖ cdif 4048 class class class wbr 5108 {copab 5173 ran crn 5659 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 distcds 17481 TarskiGcstrkg 28946 Itvcitv 28952 LineGclng 28953 hpGchpg 29292 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: plngrotlem3 29324

W3C validator