Theorem tgcgrextend 25853

Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrextend.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrextend.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrextend.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrextend.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrextend.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgcgrextend.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgcgrextend.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgcgrextend.2	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
tgcgrextend.3	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
tgcgrextend.4	⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Assertion

Ref	Expression
tgcgrextend	⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Proof of Theorem tgcgrextend

Step	Hyp	Ref	Expression
1		tgcgrextend.4	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
2	1	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
3		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
4	3	oveq1d 6939	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐶) = (𝐵 − 𝐶))
5		tkgeom.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
6		tkgeom.d	. . . . 5 ⊢ − = (dist‘𝐺)
7		tkgeom.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
8		tkgeom.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
10		tgcgrextend.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
12		tgcgrextend.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
14		tgcgrextend.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
15	14	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃)
16		tgcgrextend.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
17	16	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐸 ∈ 𝑃)
18		tgcgrextend.3	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
19	18	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
20	5, 6, 7, 9, 11, 13, 15, 17, 19, 3	tgcgreq 25850	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
21	20	oveq1d 6939	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 − 𝐹) = (𝐸 − 𝐹))
22	2, 4, 21	3eqtr4d 2824	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
23	8	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
24		tgcgrextend.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
25	24	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑃)
26	10	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
27		tgcgrextend.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
28	27	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐹 ∈ 𝑃)
29	14	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ 𝑃)
30	12	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
31	16	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ 𝑃)
32		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
33		tgcgrextend.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
34	33	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ (𝐴𝐼𝐶))
35		tgcgrextend.2	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
36	35	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (𝐷𝐼𝐹))
37	18	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
38	1	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
39	5, 6, 7, 23, 26, 29	tgcgrtriv 25852	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 − 𝐴) = (𝐷 − 𝐷))
40	5, 6, 7, 23, 26, 30, 29, 31, 37	tgcgrcomlr 25848	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
41	5, 6, 7, 23, 26, 30, 25, 29, 31, 28, 26, 29, 32, 34, 36, 37, 38, 39, 40	axtg5seg 25833	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
42	5, 6, 7, 23, 25, 26, 28, 29, 41	tgcgrcomlr 25848	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
43	22, 42	pm2.61dane 3057	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ‘cfv 6137  (class class class)co 6924  Basecbs 16266  distcds 16358  TarskiGcstrkg 25798  Itvcitv 25804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-trkgc 25816  df-trkgcb 25818  df-trkg 25821

This theorem is referenced by:  tgsegconeq  25854  tgcgrxfr  25886  lnext  25935  tgbtwnconn1lem1  25940  tgbtwnconn1lem2  25941  tgbtwnconn1lem3  25942  miriso  26038  mircgrextend  26050  midexlem  26060  opphllem  26100  flatcgra  26189  dfcgra2  26195