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Theorem krippenlem 28776

Description: Lemma for krippen 28777. 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 481	. . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐺 ∈ TarskiG)
8		krippen.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 ∈ 𝑃)
10		krippen.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐴 ∈ 𝑃)
12		krippen.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐵 ∈ 𝑃)
14		krippen.3	. . . . . . . 8 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
15	14	adantr 481	. . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))
19		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐸 = 𝐶)
20	19	oveq2d 7372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐶))
21	18, 20	breqtrd 5098	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐶))
22	3, 4, 5, 16, 7, 9, 11, 9, 13, 21	legeq 28679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐴)
23	3, 4, 5, 7, 9, 11, 9, 13, 15, 22	tgcgreq 28568	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐵)
24		krippen.5	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
25	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐵 = (𝑀‘𝐴))
26	23, 22, 25	3eqtr3rd 2783	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑋 ∈ 𝑃)
31		krippen.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝑋)
32	3, 4, 5, 27, 28, 7, 30, 31, 11	mirinv 28752	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → ((𝑀‘𝐴) = 𝐴 ↔ 𝑋 = 𝐴))
33	26, 32	mpbid 233	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑋 = 𝐴)
34		krippen.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
35	34	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐹 ∈ 𝑃)
36		krippen.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
38	37, 20	eqtr3d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝐶 − 𝐹) = (𝐶 − 𝐶))
39	3, 4, 5, 7, 9, 35, 9, 38	axtgcgrid 28549	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 = 𝐹)
40		krippen.6	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))
41	40	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐹 = (𝑁‘𝐸))
42	19, 39, 41	3eqtrrd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝑁‘𝐸) = 𝐸)
43		krippen.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
44	43	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑌 ∈ 𝑃)
45		krippen.n	. . . . . 6 ⊢ 𝑁 = (𝑆‘𝑌)
46		krippen.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
47	46	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐸 ∈ 𝑃)
48	3, 4, 5, 27, 28, 7, 44, 45, 47	mirinv 28752	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → ((𝑁‘𝐸) = 𝐸 ↔ 𝑌 = 𝐸))
49	42, 48	mpbid 233	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝑌 = 𝐸)
50	33, 49	oveq12d 7374	. . 3 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → (𝑋𝐼𝑌) = (𝐴𝐼𝐸))
51	2, 50	eleqtrrd 2842	. 2 ⊢ ((𝜑 ∧ 𝐸 = 𝐶) → 𝐶 ∈ (𝑋𝐼𝑌))
52	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐺 ∈ TarskiG)
53	52	ad2antrr 732	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐺 ∈ TarskiG)
54	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐶 ∈ 𝑃)
55		eqid 2739	. . . . . . . 8 ⊢ (𝑆‘𝐶) = (𝑆‘𝐶)
56	3, 4, 5, 27, 28, 52, 54, 55	mirf 28746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝑆‘𝐶):𝑃⟶𝑃)
57	43	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ 𝑃)
58	56, 57	ffvelcdmd 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
59	58	ad2antrr 732	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
60		simplr 774	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 ∈ 𝑃)
61	54	ad2antrr 732	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ 𝑃)
62	57	ad2antrr 732	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑌 ∈ 𝑃)
63		simprl 776	. . . . 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 28744	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝑌))
65	64	ad3antrrr 736	. . . . 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 28580	. . . 4 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐶 ∈ (𝑞𝐼𝑌))
67	29	ad3antrrr 736	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑋 ∈ 𝑃)
68	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ 𝑃)
69	68	ad2antrr 732	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐴 ∈ 𝑃)
70	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ 𝑃)
71	70	ad2antrr 732	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 ∈ 𝑃)
72		eqid 2739	. . . . . . . 8 ⊢ (𝑆‘𝑞) = (𝑆‘𝑞)
73	46	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐸 ∈ 𝑃)
74	56, 73	ffvelcdmd 7026	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
75	34	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 ∈ 𝑃)
76	56, 75	ffvelcdmd 7026	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
77	6	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
78	10	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
79	74	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
80	3, 4, 5, 77, 78, 79	tgbtwntriv1 28577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐴𝐼((𝑆‘𝐶)‘𝐸)))
81		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
82	81	oveq1d 7371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → (𝐴𝐼((𝑆‘𝐶)‘𝐸)) = (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
83	80, 82	eleqtrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 = 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
84	6	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐺 ∈ TarskiG)
85	10	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑃)
86	74	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐸) ∈ 𝑃)
87	8	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ 𝑃)
88	46	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐸 ∈ 𝑃)
89		simplr 774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐸 ≠ 𝐶)
90	3, 4, 5, 6, 10, 8, 46, 1	tgbtwncom 28574	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐸𝐼𝐴))
91	90	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝐸𝐼𝐴))
92	3, 4, 5, 27, 28, 84, 87, 55, 88	mirbtwn 28744	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐸)𝐼𝐸))
93	3, 4, 5, 84, 86, 87, 88, 92	tgbtwncom 28574	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝐸𝐼((𝑆‘𝐶)‘𝐸)))
94	3, 5, 84, 88, 87, 85, 86, 89, 91, 93	tgbtwnconn2 28662	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → (𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)) ∨ ((𝑆‘𝐶)‘𝐸) ∈ (𝐶𝐼𝐴)))
95	17	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))
96	3, 4, 5, 27, 28, 52, 54, 55, 73	mircgr 28743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐸)) = (𝐶 − 𝐸))
97	95, 96	breqtrrd 5100	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐸)))
98	97	adantr 481	. . . . . . . . . . . . . . . . 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 28680	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
100	83, 99	pm2.61dane 3021	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐸)))
101	3, 4, 5, 52, 54, 68, 74, 100	tgbtwncom 28574	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐴 ∈ (((𝑆‘𝐶)‘𝐸)𝐼𝐶))
102	6	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
103	12	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ 𝑃)
104	76	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
105	3, 4, 5, 102, 103, 104	tgbtwntriv1 28577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐵𝐼((𝑆‘𝐶)‘𝐹)))
106		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
107	106	oveq1d 7371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → (𝐵𝐼((𝑆‘𝐶)‘𝐹)) = (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
108	105, 107	eleqtrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
109	6	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
110	12	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
111	76	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐶)‘𝐹) ∈ 𝑃)
112	8	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
113	34	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ 𝑃)
114	6	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐺 ∈ TarskiG)
115	8	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐶 ∈ 𝑃)
116	46	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐸 ∈ 𝑃)
117	36	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
118		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐹 = 𝐶)
119	118	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐹) = (𝐶 − 𝐶))
120	117, 119	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐶))
121	3, 4, 5, 114, 115, 116, 115, 120	axtgcgrid 28549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐶 = 𝐸)
122	121	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝐹 = 𝐶) → 𝐸 = 𝐶)
123	122	adantlr 721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → 𝐸 = 𝐶)
124		simplr 774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → 𝐸 ≠ 𝐶)
125	124	neneqd 2939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐹 = 𝐶) → ¬ 𝐸 = 𝐶)
126	123, 125	pm2.65da 822	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ¬ 𝐹 = 𝐶)
127	126	neqned 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 ≠ 𝐶)
128	127	adantr 481	. . . . . . . . . . . . . . . . . 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 28574	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐹𝐼𝐵))
131	130	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐹𝐼𝐵))
132	3, 4, 5, 27, 28, 109, 112, 55, 113	mirbtwn 28744	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (((𝑆‘𝐶)‘𝐹)𝐼𝐹))
133	3, 4, 5, 109, 111, 112, 113, 132	tgbtwncom 28574	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐹𝐼((𝑆‘𝐶)‘𝐹)))
134	3, 5, 109, 113, 112, 110, 111, 128, 131, 133	tgbtwnconn2 28662	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → (𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)) ∨ ((𝑆‘𝐶)‘𝐹) ∈ (𝐶𝐼𝐵)))
135	17, 14, 36	3brtr3d 5103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐶 − 𝐵) ≤ (𝐶 − 𝐹))
136	135	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐵) ≤ (𝐶 − 𝐹))
137	3, 4, 5, 27, 28, 52, 54, 55, 75	mircgr 28743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐹)) = (𝐶 − 𝐹))
138	136, 137	breqtrrd 5100	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐵) ≤ (𝐶 − ((𝑆‘𝐶)‘𝐹)))
139	138	adantr 481	. . . . . . . . . . . . . . . . 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 28680	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
141	108, 140	pm2.61dane 3021	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐶)‘𝐹)))
142	3, 4, 5, 52, 54, 70, 76, 141	tgbtwncom 28574	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐶)‘𝐹)𝐼𝐶))
143	36	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
144	143, 96, 137	3eqtr4d 2784	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − ((𝑆‘𝐶)‘𝐸)) = (𝐶 − ((𝑆‘𝐶)‘𝐹)))
145	3, 4, 5, 52, 54, 74, 54, 76, 144	tgcgrcomlr 28566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐸) − 𝐶) = (((𝑆‘𝐶)‘𝐹) − 𝐶))
146	14	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
147	3, 4, 5, 52, 54, 68, 54, 70, 146	tgcgrcomlr 28566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐵 − 𝐶))
148		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆‘((𝑆‘𝐶)‘𝑌)) = (𝑆‘((𝑆‘𝐶)‘𝑌))
149	3, 4, 5, 27, 28, 52, 58, 148, 74	mircgr 28743	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸))) = (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐸)))
150		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝑌) = ((𝑆‘𝐶)‘𝑌)
151		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝐸) = ((𝑆‘𝐶)‘𝐸)
152		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆‘𝐶)‘𝐹) = ((𝑆‘𝐶)‘𝐹)
153	40	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐹 = (𝑁‘𝐸))
154	45	fveq1i 6828	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁‘𝐸) = ((𝑆‘𝑌)‘𝐸)
155	153, 154	eqtr2di 2791	. . . . . . . . . . . . . . . . . 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 28770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸)) = ((𝑆‘𝐶)‘𝐹))
157	156	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − ((𝑆‘((𝑆‘𝐶)‘𝑌))‘((𝑆‘𝐶)‘𝐸))) = (((𝑆‘𝐶)‘𝑌) − ((𝑆‘𝐶)‘𝐹)))
158	149, 157	eqtr3d 2776	. . . . . . . . . . . . . . 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 28566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝐸) − ((𝑆‘𝐶)‘𝑌)) = (((𝑆‘𝐶)‘𝐹) − ((𝑆‘𝐶)‘𝑌)))
160		eqidd 2740	. . . . . . . . . . . . . 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 28594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐶)‘𝑌)) = (𝐵 − ((𝑆‘𝐶)‘𝑌)))
162	3, 4, 5, 52, 68, 58, 70, 58, 161	tgcgrcomlr 28566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (((𝑆‘𝐶)‘𝑌) − 𝐴) = (((𝑆‘𝐶)‘𝑌) − 𝐵))
163	162	ad3antrrr 736	. . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝐺 ∈ TarskiG)
165	59	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
166	60	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝑞 ∈ 𝑃)
167	63	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → 𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶))
168		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) = 𝐶)
169	168	oveq2d 7372	. . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . 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 28552	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → ((𝑆‘𝐶)‘𝑌) = 𝑞)
172	171	oveq1d 7371	. . . . . . . . . . 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 7371	. . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) = 𝐶) → (𝑞 − 𝐴) = (𝑞 − 𝐵))
175	52	ad3antrrr 736	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐺 ∈ TarskiG)
176	58	ad3antrrr 736	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
177	54	ad3antrrr 736	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐶 ∈ 𝑃)
178	60	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝑞 ∈ 𝑃)
179		eqid 2739	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
180	68	ad3antrrr 736	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐴 ∈ 𝑃)
181	70	ad3antrrr 736	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → 𝐵 ∈ 𝑃)
182		simpr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ≠ 𝐶)
183	59	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) ∧ ((𝑆‘𝐶)‘𝑌) ≠ 𝐶) → ((𝑆‘𝐶)‘𝑌) ∈ 𝑃)
184	63	adantr 481	. . . . . . . . . . . 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 28643	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 28655	. . . . . . . . . 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 3021	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞 − 𝐴) = (𝑞 − 𝐵))
190	189	eqcomd 2745	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞 − 𝐵) = (𝑞 − 𝐴))
191		simprr 778	. . . . . . . . 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 28574	. . . . . . . 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 28745	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 = ((𝑆‘𝑞)‘𝐴))
194	193	eqcomd 2745	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → ((𝑆‘𝑞)‘𝐴) = 𝐵)
195	24	ad3antrrr 736	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝐵 = (𝑀‘𝐴))
196	31	fveq1i 6828	. . . . . . 7 ⊢ (𝑀‘𝐴) = ((𝑆‘𝑋)‘𝐴)
197	195, 196	eqtr2di 2791	. . . . . 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 28771	. . . . 5 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → 𝑞 = 𝑋)
199	198	oveq1d 7371	. . . 4 ⊢ ((((𝜑 ∧ 𝐸 ≠ 𝐶) ∧ 𝑞 ∈ 𝑃) ∧ (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵))) → (𝑞𝐼𝑌) = (𝑋𝐼𝑌))
200	66, 199	eleqtrd 2841	. . 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 28744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ ((𝑁‘𝐸)𝐼𝐸))
202	153	oveq1d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → (𝐹𝐼𝐸) = ((𝑁‘𝐸)𝐼𝐸))
203	201, 202	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ (𝐹𝐼𝐸))
204	3, 4, 5, 52, 75, 57, 73, 203	tgbtwncom 28574	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝑌 ∈ (𝐸𝐼𝐹))
205	3, 4, 5, 27, 28, 52, 54, 55, 73, 57, 75, 204	mirbtwni 28757	. . . 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 28585	. . 3 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (((𝑆‘𝐶)‘𝑌)𝐼𝐶) ∧ 𝑞 ∈ (𝐴𝐼𝐵)))
207	200, 206	r19.29a 3147	. 2 ⊢ ((𝜑 ∧ 𝐸 ≠ 𝐶) → 𝐶 ∈ (𝑋𝐼𝑌))
208	51, 207	pm2.61dane 3021	1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 distcds 17220 TarskiGcstrkg 28513 Itvcitv 28519 LineGclng 28520 cgrGccgrg 28596 ≤Gcleg 28668 pInvGcmir 28738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-s2 14801 df-s3 14802 df-trkgc 28534 df-trkgb 28535 df-trkgcb 28536 df-trkg 28539 df-cgrg 28597 df-leg 28669 df-mir 28739

This theorem is referenced by: krippen 28777

W3C validator