Theorem tgbtwnconn1lem2 27824

Description: Lemma for tgbtwnconn1 27826. (Contributed by Thierry Arnoux, 30-Apr-2019.)

Hypotheses

Ref	Expression
tgbtwnconn1.p	⊢ 𝑃 = (Base‘𝐺)
tgbtwnconn1.i	⊢ 𝐼 = (Itv‘𝐺)
tgbtwnconn1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnconn1.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnconn1.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnconn1.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnconn1.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgbtwnconn1.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
tgbtwnconn1.2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnconn1.m	⊢ − = (dist‘𝐺)
tgbtwnconn1.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgbtwnconn1.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgbtwnconn1.h	⊢ (𝜑 → 𝐻 ∈ 𝑃)
tgbtwnconn1.j	⊢ (𝜑 → 𝐽 ∈ 𝑃)
tgbtwnconn1.4	⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸))
tgbtwnconn1.5	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹))
tgbtwnconn1.6	⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻))
tgbtwnconn1.7	⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽))
tgbtwnconn1.8	⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷))
tgbtwnconn1.9	⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷))
tgbtwnconn1.10	⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶))
tgbtwnconn1.11	⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷))

Assertion

Ref	Expression
tgbtwnconn1lem2	⊢ (𝜑 → (𝐸 − 𝐹) = (𝐶 − 𝐷))

Proof of Theorem tgbtwnconn1lem2

Step	Hyp	Ref	Expression
1		tgbtwnconn1.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		tgbtwnconn1.m	. . . . 5 ⊢ − = (dist‘𝐺)
3		tgbtwnconn1.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		tgbtwnconn1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwnconn1.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
6		tgbtwnconn1.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
7	1, 2, 3, 4, 5, 6	axtgcgrrflx 27713	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐹) = (𝐹 − 𝐸))
8	7	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐸 − 𝐹) = (𝐹 − 𝐸))
9	4	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
10	5	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 ∈ 𝑃)
11		tgbtwnconn1.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ 𝑃)
12	11	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐻 ∈ 𝑃)
13		tgbtwnconn1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
14	13	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ 𝑃)
15		tgbtwnconn1.10	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶))
16	15	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐸 − 𝐻) = (𝐵 − 𝐶))
17		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
18	17	oveq1d 7424	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐶) = (𝐶 − 𝐶))
19	16, 18	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐸 − 𝐻) = (𝐶 − 𝐶))
20	1, 2, 3, 9, 10, 12, 14, 19	axtgcgrid 27714	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 = 𝐻)
21		tgbtwnconn1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
22		tgbtwnconn1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
23		tgbtwnconn1.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
24		tgbtwnconn1.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
25		tgbtwnconn1.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
26		tgbtwnconn1.3	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
27		tgbtwnconn1.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ 𝑃)
28		tgbtwnconn1.4	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸))
29		tgbtwnconn1.5	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹))
30		tgbtwnconn1.6	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻))
31		tgbtwnconn1.7	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽))
32		tgbtwnconn1.8	. . . . . . . 8 ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷))
33		tgbtwnconn1.9	. . . . . . . 8 ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷))
34		tgbtwnconn1.11	. . . . . . . 8 ⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷))
35	1, 3, 4, 21, 22, 13, 23, 24, 25, 26, 2, 5, 6, 11, 27, 28, 29, 30, 31, 32, 33, 15, 34	tgbtwnconn1lem1 27823	. . . . . . 7 ⊢ (𝜑 → 𝐻 = 𝐽)
36	35	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐻 = 𝐽)
37	20, 36	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 = 𝐽)
38	37	oveq2d 7425	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐹 − 𝐸) = (𝐹 − 𝐽))
39	34	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐹 − 𝐽) = (𝐵 − 𝐷))
40	17	oveq1d 7424	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐷) = (𝐶 − 𝐷))
41	38, 39, 40	3eqtrd 2777	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐹 − 𝐸) = (𝐶 − 𝐷))
42	8, 41	eqtrd 2773	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐸 − 𝐹) = (𝐶 − 𝐷))
43	4	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
44	6	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ 𝑃)
45	5	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ 𝑃)
46	23	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
47	13	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
48	22	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
49	27	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐽 ∈ 𝑃)
50		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
51	1, 2, 3, 4, 21, 22, 13, 6, 25, 29	tgbtwnexch3 27745	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))
52	51	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐹))
53	35	oveq2d 7425	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝐼𝐻) = (𝐴𝐼𝐽))
54	30, 53	eleqtrd 2836	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐽))
55	1, 2, 3, 4, 21, 23, 5, 27, 28, 54	tgbtwnexch3 27745	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐽))
56	1, 2, 3, 4, 23, 5, 27, 55	tgbtwncom 27739	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐽𝐼𝐷))
57	56	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (𝐽𝐼𝐷))
58	35	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐻 = 𝐽)
59	58	oveq2d 7425	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − 𝐻) = (𝐸 − 𝐽))
60	15	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − 𝐻) = (𝐵 − 𝐶))
61	1, 2, 3, 43, 45, 49	axtgcgrrflx 27713	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − 𝐽) = (𝐽 − 𝐸))
62	59, 60, 61	3eqtr3d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) = (𝐽 − 𝐸))
63	33, 32	eqtr4d 2776	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐸 − 𝐷))
64	63	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐹) = (𝐸 − 𝐷))
65	1, 2, 3, 4, 21, 22, 23, 5, 26, 28	tgbtwnexch3 27745	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐸))
66	65	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ (𝐵𝐼𝐸))
67	1, 2, 3, 4, 21, 13, 6, 27, 29, 31	tgbtwnexch3 27745	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶𝐼𝐽))
68	1, 2, 3, 4, 13, 6, 27, 67	tgbtwncom 27739	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽𝐼𝐶))
69	68	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ (𝐽𝐼𝐶))
70	1, 2, 3, 4, 27, 6	axtgcgrrflx 27713	. . . . . . 7 ⊢ (𝜑 → (𝐽 − 𝐹) = (𝐹 − 𝐽))
71	70, 34	eqtr2d 2774	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐽 − 𝐹))
72	71	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐷) = (𝐽 − 𝐹))
73	1, 2, 3, 4, 13, 6, 5, 23, 63	tgcgrcomlr 27731	. . . . . . 7 ⊢ (𝜑 → (𝐹 − 𝐶) = (𝐷 − 𝐸))
74	73	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐹 − 𝐶) = (𝐷 − 𝐸))
75	74	eqcomd 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐷 − 𝐸) = (𝐹 − 𝐶))
76	1, 2, 3, 43, 48, 46, 45, 49, 44, 47, 66, 69, 72, 75	tgcgrextend 27736	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐸) = (𝐽 − 𝐶))
77	1, 2, 3, 43, 47, 45	axtgcgrrflx 27713	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐸) = (𝐸 − 𝐶))
78	1, 2, 3, 43, 48, 47, 44, 49, 45, 46, 45, 47, 50, 52, 57, 62, 64, 76, 77	axtg5seg 27716	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐹 − 𝐸) = (𝐷 − 𝐶))
79	1, 2, 3, 43, 44, 45, 46, 47, 78	tgcgrcomlr 27731	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − 𝐹) = (𝐶 − 𝐷))
80	42, 79	pm2.61dane 3030	1 ⊢ (𝜑 → (𝐸 − 𝐹) = (𝐶 − 𝐷))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiGcstrkg 27678  Itvcitv 27684

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-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-trkgc 27699  df-trkgb 27700  df-trkgcb 27701  df-trkg 27704

This theorem is referenced by:  tgbtwnconn1lem3  27825