Theorem tgbtwnconn1lem1 28627

Description: Lemma for tgbtwnconn1 28630. (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
tgbtwnconn1lem1	⊢ (𝜑 → 𝐻 = 𝐽)

Proof of Theorem tgbtwnconn1lem1

Step	Hyp	Ref	Expression
1		tgbtwnconn1.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tgbtwnconn1.m	. 2 ⊢ − = (dist‘𝐺)
3		tgbtwnconn1.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		tgbtwnconn1.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwnconn1.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
6		tgbtwnconn1.j	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝑃)
7		tgbtwnconn1.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		tgbtwnconn1.h	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑃)
9		tgbtwnconn1.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
10		tgbtwnconn1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
11		tgbtwnconn1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
12		tgbtwnconn1.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
13		tgbtwnconn1.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸))
14	1, 2, 3, 4, 7, 5, 11, 10, 12, 13	tgbtwnexch 28553	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐸))
15		tgbtwnconn1.6	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻))
16	1, 2, 3, 4, 7, 5, 10, 8, 14, 15	tgbtwnexch 28553	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐻))
17		tgbtwnconn1.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
18		tgbtwnconn1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
19		tgbtwnconn1.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
20		tgbtwnconn1.5	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹))
21	1, 2, 3, 4, 7, 5, 18, 17, 19, 20	tgbtwnexch 28553	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐹))
22		tgbtwnconn1.7	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽))
23	1, 2, 3, 4, 7, 5, 17, 6, 21, 22	tgbtwnexch 28553	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐽))
24	1, 2, 3, 4, 7, 5, 10, 8, 14, 15	tgbtwnexch3 28549	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐵𝐼𝐻))
25	1, 2, 3, 4, 7, 18, 17, 6, 20, 22	tgbtwnexch 28553	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐽))
26	1, 2, 3, 4, 7, 5, 18, 6, 19, 25	tgbtwnexch3 28549	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐽))
27	1, 2, 3, 4, 5, 18, 6, 26	tgbtwncom 28543	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐽𝐼𝐵))
28	1, 2, 3, 4, 7, 5, 11, 10, 12, 13	tgbtwnexch3 28549	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐸))
29	1, 2, 3, 4, 7, 18, 17, 6, 20, 22	tgbtwnexch3 28549	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶𝐼𝐽))
30	1, 2, 3, 4, 18, 17, 6, 29	tgbtwncom 28543	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽𝐼𝐶))
31	1, 2, 3, 4, 6, 17	axtgcgrrflx 28517	. . . . 5 ⊢ (𝜑 → (𝐽 − 𝐹) = (𝐹 − 𝐽))
32		tgbtwnconn1.11	. . . . 5 ⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷))
33	31, 32	eqtr2d 2773	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐽 − 𝐹))
34		tgbtwnconn1.8	. . . . . 6 ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷))
35		tgbtwnconn1.9	. . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷))
36	34, 35	eqtr4d 2775	. . . . 5 ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐹))
37	1, 2, 3, 4, 10, 11, 18, 17, 36	tgcgrcomlr 28535	. . . 4 ⊢ (𝜑 → (𝐷 − 𝐸) = (𝐹 − 𝐶))
38	1, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37	tgcgrextend 28540	. . 3 ⊢ (𝜑 → (𝐵 − 𝐸) = (𝐽 − 𝐶))
39		tgbtwnconn1.10	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶))
40	1, 2, 3, 4, 10, 8, 5, 18, 39	tgcgrcomr 28533	. . 3 ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐶 − 𝐵))
41	1, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 40	tgcgrextend 28540	. 2 ⊢ (𝜑 → (𝐵 − 𝐻) = (𝐽 − 𝐵))
42	1, 2, 3, 4, 5, 6	axtgcgrrflx 28517	. 2 ⊢ (𝜑 → (𝐵 − 𝐽) = (𝐽 − 𝐵))
43	1, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 41, 42	tgsegconeq 28541	1 ⊢ (𝜑 → 𝐻 = 𝐽)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  distcds 17190  TarskiGcstrkg 28482  Itvcitv 28488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-trkgc 28503  df-trkgb 28504  df-trkgcb 28505  df-trkg 28508

This theorem is referenced by:  tgbtwnconn1lem2  28628  tgbtwnconn1lem3  28629