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Theorem plngrotlem2 28992

Description: Lemma for plngrot 28994. (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 736	. . . . 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 736	. . . . 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 736	. . . . 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 736	. . . . 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 736	. . . . 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 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑊 ∈ 𝑃)
18		plngrotlem2.2	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))
19	18	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑌 ∈ (𝑍𝐼𝑊))
20		plngrotlem2.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
21	20	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑌 ≠ 𝑊)
22		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍))
23	22	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍))
24		simpr 488	. . . . 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 28991	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
26	5	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝐺 ∈ TarskiG)
27	7	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
28	11	eldifad 3916	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
29	28	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ∈ 𝑃)
30	9	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ∈ 𝑃)
31	7	eldifad 3916	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
32	1, 2, 3, 5, 31, 9, 13	tglinerflx2 28800	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐿𝑌))
33		elndif 4086	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (𝑋𝐿𝑌) → ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
35		nelne2 3055	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ ¬ 𝑌 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) → 𝑍 ≠ 𝑌)
36	11, 34, 35	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
37	36	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ≠ 𝑌)
38	1, 2, 3, 26, 29, 30, 37	tglinecom 28801	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
39	16	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑊 ∈ 𝑃)
40	20	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ≠ 𝑊)
41	1, 2, 3, 5, 9, 16, 28, 20, 18	btwnlng2 28786	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑊))
42	41	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑍 ∈ (𝑌𝐿𝑊))
43	1, 2, 3, 26, 30, 39, 40, 29, 37, 42	tglineelsb2 28798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑌𝐿𝑊) = (𝑌𝐿𝑍))
44	1, 2, 3, 26, 30, 39, 40	tglinecom 28801	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑌𝐿𝑊) = (𝑊𝐿𝑌))
45	38, 43, 44	3eqtr2d 2803	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑍𝐿𝑌) = (𝑊𝐿𝑌))
46	45	difeq2d 4080	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑃 ∖ (𝑍𝐿𝑌)) = (𝑃 ∖ (𝑊𝐿𝑌)))
47	27, 46	eleqtrd 2864	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ∈ (𝑃 ∖ (𝑊𝐿𝑌)))
48	20	neneqd 2962	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑌 = 𝑊)
49	5	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝐺 ∈ TarskiG)
50	31	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑋 ∈ 𝑃)
51	9	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ 𝑃)
52	13	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑋 ≠ 𝑌)
53	1, 2, 3, 49, 50, 51, 52	tgelrnln 28796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
54	28	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ∈ 𝑃)
55	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ≠ 𝑌)
56	1, 2, 3, 49, 54, 51, 55	tgelrnln 28796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ∈ ran 𝐿)
57	1, 2, 3, 5, 28, 9, 36	tglinerflx1 28799	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑌))
58	57	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑍 ∈ (𝑍𝐿𝑌))
59	11	eldifbd 3917	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
60	59	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
61		nelne1 3054	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ (𝑍𝐿𝑌) ∧ ¬ 𝑍 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ≠ (𝑋𝐿𝑌))
62	58, 60, 61	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑍𝐿𝑌) ≠ (𝑋𝐿𝑌))
63	62	necomd 3012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑌))
64	32	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑋𝐿𝑌))
65	1, 2, 3, 49, 54, 51, 55	tglinerflx2 28800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑍𝐿𝑌))
66	64, 65	elind 4152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑌)))
67		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ (𝑋𝐿𝑌))
68	16	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ 𝑃)
69	18	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 ∈ (𝑍𝐼𝑊))
70	1, 2, 3, 49, 54, 51, 68, 55, 69	btwnlng3 28787	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ (𝑍𝐿𝑌))
71	67, 70	elind 4152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑊 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑌)))
72	1, 2, 3, 49, 53, 56, 63, 66, 71	tglineineq 28809	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝑋𝐿𝑌)) → 𝑌 = 𝑊)
73	48, 72	mtand 825	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
74	73	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
75	39, 74	eldifd 3915	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑊 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
76	13	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑋 ≠ 𝑌)
77		eqid 2762	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
78	1, 77, 2, 5, 28, 9, 16, 18	tgbtwncom 28654	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑊𝐼𝑍))
79	78	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ∈ (𝑊𝐼𝑍))
80	37	necomd 3012	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑌 ≠ 𝑍)
81	1, 77, 2, 15, 16, 28, 32, 73, 59, 78	islnoppd 28910	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊𝑂𝑍)
82	81	ad3antrrr 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑊𝑂𝑍)
83	5	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝐺 ∈ TarskiG)
84	83	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝐺 ∈ TarskiG)
85	31	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑋 ∈ 𝑃)
86	9	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑌 ∈ 𝑃)
87	13	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑋 ≠ 𝑌)
88	1, 2, 3, 83, 85, 86, 87	tgelrnln 28796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
89	88	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑋𝐿𝑌) ∈ ran 𝐿)
90	11	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
91	1, 2, 3, 4, 83, 88, 90, 22	plngssp 28985	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ 𝑃)
92	91	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠 ∈ 𝑃)
93	16	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑊 ∈ 𝑃)
94	28	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍 ∈ 𝑃)
95		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠𝑂𝑍)
96	1, 2, 3, 15, 84, 89, 92, 93, 94, 95	lnopp2hpgb 28933	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑍 ↔ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
97	82, 96	mpbid 234	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
98	97	orcd 884	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))
99	98	ex 416	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → (𝑠𝑂𝑍 → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
100	99	ex 416	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠𝑂𝑍 → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
101	100	a2d 29	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
102		df-or 859	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍))
103		df-or 859	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
104	101, 102, 103	3imtr4g 298	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))))
105	104	imp 410	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
106		3orass 1101	. . . . . . . 8 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
107	105, 106	sylibr 236	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊))
108	88	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑋𝐿𝑌) ∈ ran 𝐿)
109	91	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ 𝑃)
110	1, 2, 3, 4, 26, 108, 75, 15, 109	elplng 28984	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑊) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊)))
111	107, 110	mpbird 259	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑊))
112	32	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑌 ∈ (𝑋𝐿𝑌))
113	59	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
114	73	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑊 ∈ (𝑋𝐿𝑌))
115	18	ad3antrrr 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑌 ∈ (𝑍𝐼𝑊))
116	1, 77, 2, 15, 94, 93, 112, 113, 114, 115	islnoppd 28910	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍𝑂𝑊)
117	1, 2, 3, 15, 84, 89, 94, 93, 116	lnoppnhpg 28934	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
118	89	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → (𝑋𝐿𝑌) ∈ ran 𝐿)
119	84	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝐺 ∈ TarskiG)
120	93	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑊 ∈ 𝑃)
121	92	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑠 ∈ 𝑃)
122		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑊𝑂𝑠)
123	1, 77, 2, 15, 3, 118, 119, 120, 121, 122	oppcom 28914	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑊𝑂𝑠) → 𝑠𝑂𝑊)
124	89	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → (𝑋𝐿𝑌) ∈ ran 𝐿)
125	84	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝐺 ∈ TarskiG)
126	92	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑠 ∈ 𝑃)
127	93	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑊 ∈ 𝑃)
128		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑠𝑂𝑊)
129	1, 77, 2, 15, 3, 124, 125, 126, 127, 128	oppcom 28914	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) ∧ 𝑠𝑂𝑊) → 𝑊𝑂𝑠)
130	123, 129	impbida 810	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑠 ↔ 𝑠𝑂𝑊))
131	1, 77, 2, 15, 3, 89, 84, 92, 94, 95	oppcom 28914	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → 𝑍𝑂𝑠)
132	1, 2, 3, 15, 84, 89, 94, 93, 92, 131	lnopp2hpgb 28933	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑊𝑂𝑠 ↔ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
133	130, 132	bitr3d 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → (𝑠𝑂𝑊 ↔ 𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
134	117, 133	mtbird 327	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ 𝑠𝑂𝑍) → ¬ 𝑠𝑂𝑊)
135	134	ex 416	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → (𝑠𝑂𝑍 → ¬ 𝑠𝑂𝑊))
136	135	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → (𝑠𝑂𝑍 → ¬ 𝑠𝑂𝑊)))
137	136	a2d 29	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊)))
138	137	imp 410	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊))
139	102, 138	sylan2b 603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → ¬ 𝑠𝑂𝑊))
140	139	imp 410	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → ¬ 𝑠𝑂𝑊)
141		df-3or 1099	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ∨ 𝑠𝑂𝑊))
142		orcom 881	. . . . . . . . . . . . . . 15 ⊢ (((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ∨ 𝑠𝑂𝑊) ↔ (𝑠𝑂𝑊 ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
143		df-or 859	. . . . . . . . . . . . . . 15 ⊢ ((𝑠𝑂𝑊 ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)) ↔ (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
144	141, 142, 143	3bitri 299	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∨ 𝑠𝑂𝑊) ↔ (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
145	107, 144	sylib 220	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠𝑂𝑊 → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)))
146	145	imp 410	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
147		df-or 859	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊) ↔ (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
148	146, 147	sylib 220	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
149	148	imp 410	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠𝑂𝑊) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
150	149	an32s 662	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) ∧ ¬ 𝑠𝑂𝑊) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
151	140, 150	mpdan 697	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) ∧ ¬ 𝑠 ∈ (𝑋𝐿𝑌)) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊)
152	151	ex 416	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (¬ 𝑠 ∈ (𝑋𝐿𝑌) → 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊))
153	152, 147	sylibr 236	. . . . . 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 28991	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑊𝐿𝑌)𝐸𝑋))
155	45	eqcomd 2768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → (𝑊𝐿𝑌) = (𝑍𝐿𝑌))
156	155	oveq1d 7411	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → ((𝑊𝐿𝑌)𝐸𝑋) = ((𝑍𝐿𝑌)𝐸𝑋))
157	154, 156	eleqtrd 2864	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) ∧ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
158	1, 2, 3, 4, 83, 88, 90, 15, 91	elplng 28984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)))
159	22, 158	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍))
160		3orass 1101	. . . . . 6 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍) ↔ (𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)))
161		orordi 939	. . . . . 6 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ (𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍)) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
162	160, 161	bitri 277	. . . . 5 ⊢ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍 ∨ 𝑠𝑂𝑍) ↔ ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
163	159, 162	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → ((𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍) ∨ (𝑠 ∈ (𝑋𝐿𝑌) ∨ 𝑠𝑂𝑍)))
164	25, 157, 163	mpjaodan 971	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
165	164	ex 416	. 2 ⊢ (𝜑 → (𝑠 ∈ ((𝑋𝐿𝑌)𝐸𝑍) → 𝑠 ∈ ((𝑍𝐿𝑌)𝐸𝑋)))
166	165	ssrdv 3942	1 ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 858 ∨ w3o 1097 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∃wrex 3086 ∖ cdif 3901 ⊆ wss 3904 class class class wbr 5100 {copab 5162 ran crn 5648 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 distcds 17295 TarskiGcstrkg 28593 Itvcitv 28599 LineGclng 28600 hpGchpg 28927 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: plngrotlem3 28993

W3C validator