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Theorem krippenlem 27941

Description: Lemma for krippen 27942. We can assume krippen.7 "without loss of generality". (Contributed by Thierry Arnoux, 12-Aug-2019.)

**Hypotheses**
Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
krippen.m	⊢ 𝑀 = (𝑆‘𝑋)
krippen.n	⊢ 𝑁 = (𝑆‘𝑌)
krippen.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
krippen.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
krippen.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
krippen.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
krippen.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
krippen.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
krippen.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
krippen.1	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
krippen.2	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))
krippen.3	⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
krippen.4	⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))
krippen.5	⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
krippen.6	⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))
krippen.l	⊢ ≤ = (≤G‘𝐺)
krippen.7	⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))

**Assertion**
Ref	Expression
krippenlem	⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

Proof of Theorem krippenlem Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		krippen.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
2	1	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐸))
3		mirval.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
4		mirval.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
5		mirval.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
6		mirval.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐺 ∈ TarskiG)
8		krippen.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9	8	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 ∈ 𝑃)
10		krippen.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐴 ∈ 𝑃)
12		krippen.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐵 ∈ 𝑃)
14		krippen.3	. . . . . . . 8 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
15	14	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
16		krippen.l	. . . . . . . 8 ⊢ ≤ = (≤G‘𝐺)
17		krippen.7	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))
18	17	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))
19		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐸 = 𝐶)
20	19	oveq2d 7425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐶))
21	18, 20	breqtrd 5175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐶))
22	3, 4, 5, 16, 7, 9, 11, 9, 13, 21	legeq 27844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐴)
23	3, 4, 5, 7, 9, 11, 9, 13, 15, 22	tgcgreq 27733	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐵)
24		krippen.5	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
25	24	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐵 = (𝑀‘𝐴))
26	23, 22, 25	3eqtr3rd 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝑀‘𝐴) = 𝐴)
27		mirval.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
28		mirval.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
29		krippen.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
30	29	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑋 ∈ 𝑃)
31		krippen.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝑋)
32	3, 4, 5, 27, 28, 7, 30, 31, 11	mirinv 27917	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → ((𝑀‘𝐴) = 𝐴 ↔ 𝑋 = 𝐴))
33	26, 32	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑋 = 𝐴)
34		krippen.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
35	34	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐹 ∈ 𝑃)
36		krippen.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))
37	36	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
38	37, 20	eqtr3d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐹) = (𝐶 − 𝐶))
39	3, 4, 5, 7, 9, 35, 9, 38	axtgcgrid 27714	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐹)
40		krippen.6	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))
41	40	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐹 = (𝑁‘𝐸))
42	19, 39, 41	3eqtrrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝑁‘𝐸) = 𝐸)
43		krippen.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
44	43	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑌 ∈ 𝑃)
45		krippen.n	. . . . . 6 ⊢ 𝑁 = (𝑆‘𝑌)
46		krippen.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
47	46	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐸 ∈ 𝑃)
48	3, 4, 5, 27, 28, 7, 44, 45, 47	mirinv 27917	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → ((𝑁‘𝐸) = 𝐸 ↔ 𝑌 = 𝐸))
49	42, 48	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑌 = 𝐸)
50	33, 49	oveq12d 7427	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝑋𝐼𝑌) = (𝐴𝐼𝐸))
51	2, 50	eleqtrrd 2837	. 2 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 ∈ (𝑋𝐼𝑌))
52	6	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐺 ∈ TarskiG)
53	52	ad2antrr 725	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐺 ∈ TarskiG)
54	8	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐶 ∈ 𝑃)
55		eqid 2733	. . . . . . . 8 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶)
56	3, 4, 5, 27, 28, 52, 54, 55	mirf 27911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝑆‘𝐶):𝑃⟶𝑃)
57	43	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ 𝑃)
58	56, 57	ffvelcdmd 7088	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
59	58	ad2antrr 725	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
60		simplr 768	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 ∈ 𝑃)
61	54	ad2antrr 725	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ 𝑃)
62	57	ad2antrr 725	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑌 ∈ 𝑃)
63		simprl 770	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶))
64	3, 4, 5, 27, 28, 6, 8, 55, 43	mirbtwn 27909	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝑌))
65	64	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝑌))
66	3, 4, 5, 53, 59, 60, 61, 62, 63, 65	tgbtwnexch3 27745	. . . 4 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ (𝑞𝐼𝑌))
67	29	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑋 ∈ 𝑃)
68	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ 𝑃)
69	68	ad2antrr 725	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐴 ∈ 𝑃)
70	12	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ 𝑃)
71	70	ad2antrr 725	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 ∈ 𝑃)
72		eqid 2733	. . . . . . . 8 ⊢ (𝑆‘𝑞) = (𝑆‘𝑞)
73	46	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐸 ∈ 𝑃)
74	56, 73	ffvelcdmd 7088	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
75	34	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 ∈ 𝑃)
76	56, 75	ffvelcdmd 7088	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
77	6	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
78	10	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
79	74	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
80	3, 4, 5, 77, 78, 79	tgbtwntriv1 27742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐸)))
81		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
82	81	oveq1d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → (𝐴𝐼((𝑆‘𝐶)‘𝐸)) = (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
83	80, 82	eleqtrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
84	6	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐺 ∈ TarskiG)
85	10	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑃)
86	74	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
87	8	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ 𝑃)
88	46	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐸 ∈ 𝑃)
89		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐸 ≠ 𝐶)
90	3, 4, 5, 6, 10, 8, 46, 1	tgbtwncom 27739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐸𝐼𝐴))
91	90	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝐸𝐼𝐴))
92	3, 4, 5, 27, 28, 84, 87, 55, 88	mirbtwn 27909	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐸)𝐼𝐸))
93	3, 4, 5, 84, 86, 87, 88, 92	tgbtwncom 27739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝐸𝐼((𝑆‘𝐶)‘𝐸)))
94	3, 5, 84, 88, 87, 85, 86, 89, 91, 93	tgbtwnconn2 27827	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → (𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)) ∨ ((𝑆‘𝐶)‘𝐸) ∈ (𝐶𝐼𝐴)))
95	17	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))
96	3, 4, 5, 27, 28, 52, 54, 55, 73	mircgr 27908	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐸)) = (𝐶 − 𝐸))
97	95, 96	breqtrrd 5177	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐸)))
98	97	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐸)))
99	3, 4, 5, 16, 84, 85, 86, 87, 85, 94, 98	legbtwn 27845	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
100	83, 99	pm2.61dane 3030	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
101	3, 4, 5, 52, 54, 68, 74, 100	tgbtwncom 27739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ (((𝑆‘𝐶)‘𝐸)𝐼𝐶))
102	6	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
103	12	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ 𝑃)
104	76	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
105	3, 4, 5, 102, 103, 104	tgbtwntriv1 27742	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐵𝐼((𝑆‘𝐶)‘𝐹)))
106		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
107	106	oveq1d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → (𝐵𝐼((𝑆‘𝐶)‘𝐹)) = (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
108	105, 107	eleqtrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
109	6	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
110	12	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
111	76	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
112	8	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
113	34	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ 𝑃)
114	6	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐺 ∈ TarskiG)
115	8	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐶 ∈ 𝑃)
116	46	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐸 ∈ 𝑃)
117	36	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
118		simpr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐹 = 𝐶)
119	118	oveq2d 7425	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐹) = (𝐶 − 𝐶))
120	117, 119	eqtrd 2773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐶))
121	3, 4, 5, 114, 115, 116, 115, 120	axtgcgrid 27714	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐶 = 𝐸)
122	121	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐸 = 𝐶)
123	122	adantlr 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → 𝐸 = 𝐶)
124		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → 𝐸 ≠ 𝐶)
125	124	neneqd 2946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → ¬ 𝐸 = 𝐶)
126	123, 125	pm2.65da 816	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ¬ 𝐹 = 𝐶)
127	126	neqned 2948	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 ≠ 𝐶)
128	127	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐹 ≠ 𝐶)
129		krippen.2	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))
130	3, 4, 5, 6, 12, 8, 34, 129	tgbtwncom 27739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐹𝐼𝐵))
131	130	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐹𝐼𝐵))
132	3, 4, 5, 27, 28, 109, 112, 55, 113	mirbtwn 27909	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐹)𝐼𝐹))
133	3, 4, 5, 109, 111, 112, 113, 132	tgbtwncom 27739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐹𝐼((𝑆‘𝐶)‘𝐹)))
134	3, 5, 109, 113, 112, 110, 111, 128, 131, 133	tgbtwnconn2 27827	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → (𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)) ∨ ((𝑆‘𝐶)‘𝐹) ∈ (𝐶𝐼𝐵)))
135	17, 14, 36	3brtr3d 5180	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐶 − 𝐵) ≤ (𝐶 − 𝐹))
136	135	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐵) ≤ (𝐶 − 𝐹))
137	3, 4, 5, 27, 28, 52, 54, 55, 75	mircgr 27908	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐹)) = (𝐶 − 𝐹))
138	136, 137	breqtrrd 5177	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐵) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐹)))
139	138	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐵) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐹)))
140	3, 4, 5, 16, 109, 110, 111, 112, 110, 134, 139	legbtwn 27845	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
141	108, 140	pm2.61dane 3030	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
142	3, 4, 5, 52, 54, 70, 76, 141	tgbtwncom 27739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐶)‘𝐹)𝐼𝐶))
143	36	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
144	143, 96, 137	3eqtr4d 2783	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐸)) = (𝐶 − ((𝑆‘𝐶)‘𝐹)))
145	3, 4, 5, 52, 54, 74, 54, 76, 144	tgcgrcomlr 27731	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐸) − 𝐶) = (((𝑆‘𝐶)‘𝐹) − 𝐶))
146	14	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
147	3, 4, 5, 52, 54, 68, 54, 70, 146	tgcgrcomlr 27731	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐵 − 𝐶))
148		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆‘((𝑆‘𝐶)‘𝑌)) = (𝑆‘((𝑆‘𝐶)‘𝑌))
149	3, 4, 5, 27, 28, 52, 58, 148, 74	mircgr 27908	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸))) = (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐸)))
150		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝑌) = ((𝑆‘𝐶)‘𝑌)
151		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝐸) = ((𝑆‘𝐶)‘𝐸)
152		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝐹) = ((𝑆‘𝐶)‘𝐹)
153	40	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 = (𝑁‘𝐸))
154	45	fveq1i 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁‘𝐸) = ((𝑆‘𝑌)‘𝐸)
155	153, 154	eqtr2di 2790	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝑌)‘𝐸) = 𝐹)
156	3, 4, 5, 27, 28, 52, 55, 150, 151, 152, 54, 57, 73, 75, 155	mirauto 27935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸)) = ((𝑆‘𝐶)‘𝐹))
157	156	oveq2d 7425	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸))) = (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐹)))
158	149, 157	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐸)) = (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐹)))
159	3, 4, 5, 52, 58, 74, 58, 76, 158	tgcgrcomlr 27731	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐸) − ((𝑆‘𝐶)‘𝑌)) = (((𝑆‘𝐶)‘𝐹) − ((𝑆‘𝐶)‘𝑌)))
160		eqidd 2734	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝑌)) = (𝐶 − ((𝑆‘𝐶)‘𝑌)))
161	3, 4, 5, 52, 74, 68, 54, 58, 76, 70, 54, 58, 101, 142, 145, 147, 159, 160	tgifscgr 27759	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐶)‘𝑌)) = (𝐵 − ((𝑆‘𝐶)‘𝑌)))
162	3, 4, 5, 52, 68, 58, 70, 58, 161	tgcgrcomlr 27731	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐴) = (((𝑆‘𝐶)‘𝑌) − 𝐵))
163	162	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐴) = (((𝑆‘𝐶)‘𝑌) − 𝐵))
164	53	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝐺 ∈ TarskiG)
165	59	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
166	60	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝑞 ∈ 𝑃)
167	63	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶))
168		simpr 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) = 𝐶)
169	168	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (((𝑆‘𝐶)‘𝑌)𝐼((𝑆‘𝐶)‘𝑌)) = (((𝑆‘𝐶)‘𝑌)𝐼𝐶))
170	167, 169	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼((𝑆‘𝐶)‘𝑌)))
171	3, 4, 5, 164, 165, 166, 170	axtgbtwnid 27717	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) = 𝑞)
172	171	oveq1d 7424	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐴) = (𝑞 − 𝐴))
173	171	oveq1d 7424	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐵) = (𝑞 − 𝐵))
174	163, 172, 173	3eqtr3d 2781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (𝑞 − 𝐴) = (𝑞 − 𝐵))
175	52	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐺 ∈ TarskiG)
176	58	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
177	54	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐶 ∈ 𝑃)
178	60	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝑞 ∈ 𝑃)
179		eqid 2733	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
180	68	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐴 ∈ 𝑃)
181	70	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐵 ∈ 𝑃)
182		simpr 486	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ≠ 𝐶)
183	59	adantr 482	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
184	63	adantr 482	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶))
185	3, 27, 5, 175, 183, 178, 177, 184	btwncolg3 27808	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → (𝐶 ∈ (((𝑆‘𝐶)‘𝑌)𝐿𝑞) ∨ ((𝑆‘𝐶)‘𝑌) = 𝑞))
186	162	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐴) = (((𝑆‘𝐶)‘𝑌) − 𝐵))
187	146	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
188	3, 27, 5, 175, 176, 177, 178, 179, 180, 181, 4, 182, 185, 186, 187	lncgr 27820	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → (𝑞 − 𝐴) = (𝑞 − 𝐵))
189	174, 188	pm2.61dane 3030	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞 − 𝐴) = (𝑞 − 𝐵))
190	189	eqcomd 2739	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞 − 𝐵) = (𝑞 − 𝐴))
191		simprr 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 ∈ (𝐴𝐼𝐵))
192	3, 4, 5, 53, 69, 60, 71, 191	tgbtwncom 27739	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 ∈ (𝐵𝐼𝐴))
193	3, 4, 5, 27, 28, 53, 60, 72, 69, 71, 190, 192	ismir 27910	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 = ((𝑆‘𝑞)‘𝐴))
194	193	eqcomd 2739	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → ((𝑆‘𝑞)‘𝐴) = 𝐵)
195	24	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 = (𝑀‘𝐴))
196	31	fveq1i 6893	. . . . . . 7 ⊢ (𝑀‘𝐴) = ((𝑆‘𝑋)‘𝐴)
197	195, 196	eqtr2di 2790	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → ((𝑆‘𝑋)‘𝐴) = 𝐵)
198	3, 4, 5, 27, 28, 53, 60, 67, 69, 71, 194, 197	miduniq 27936	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 = 𝑋)
199	198	oveq1d 7424	. . . 4 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞𝐼𝑌) = (𝑋𝐼𝑌))
200	66, 199	eleqtrd 2836	. . 3 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ (𝑋𝐼𝑌))
201	3, 4, 5, 27, 28, 52, 57, 45, 73	mirbtwn 27909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ ((𝑁‘𝐸)𝐼𝐸))
202	153	oveq1d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐹𝐼𝐸) = ((𝑁‘𝐸)𝐼𝐸))
203	201, 202	eleqtrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ (𝐹𝐼𝐸))
204	3, 4, 5, 52, 75, 57, 73, 203	tgbtwncom 27739	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ (𝐸𝐼𝐹))
205	3, 4, 5, 27, 28, 52, 54, 55, 73, 57, 75, 204	mirbtwni 27922	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ (((𝑆‘𝐶)‘𝐸)𝐼((𝑆‘𝐶)‘𝐹)))
206	3, 4, 5, 52, 74, 68, 54, 76, 70, 58, 101, 142, 205	tgtrisegint 27750	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵)))
207	200, 206	r19.29a 3163	. 2 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐶 ∈ (𝑋𝐼𝑌))
208	51, 207	pm2.61dane 3030	1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 TarskiGcstrkg 27678 Itvcitv 27684 LineGclng 27685 cgrGccgrg 27761 ≤Gcleg 27833 pInvGcmir 27903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-concat 14521 df-s1 14546 df-s2 14799 df-s3 14800 df-trkgc 27699 df-trkgb 27700 df-trkgcb 27701 df-trkg 27704 df-cgrg 27762 df-leg 27834 df-mir 27904

This theorem is referenced by: krippen 27942

W3C validator