Step | Hyp | Ref
| Expression |
1 | | trgcgrg.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
2 | | trgcgrg.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
3 | | trgcgrg.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
4 | 1, 2, 3 | s3cld 13617 |
. . . . . 6
⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑃) |
5 | | wrdf 13310 |
. . . . . 6
⊢
(〈“𝐴𝐵𝐶”〉 ∈ Word 𝑃 → 〈“𝐴𝐵𝐶”〉:(0..^(#‘〈“𝐴𝐵𝐶”〉))⟶𝑃) |
6 | 4, 5 | syl 17 |
. . . . 5
⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉:(0..^(#‘〈“𝐴𝐵𝐶”〉))⟶𝑃) |
7 | | s3len 13639 |
. . . . . . . 8
⊢
(#‘〈“𝐴𝐵𝐶”〉) = 3 |
8 | 7 | oveq2i 6661 |
. . . . . . 7
⊢
(0..^(#‘〈“𝐴𝐵𝐶”〉)) = (0..^3) |
9 | | fzo0to3tp 12554 |
. . . . . . 7
⊢ (0..^3) =
{0, 1, 2} |
10 | 8, 9 | eqtri 2644 |
. . . . . 6
⊢
(0..^(#‘〈“𝐴𝐵𝐶”〉)) = {0, 1, 2} |
11 | 10 | feq2i 6037 |
. . . . 5
⊢
(〈“𝐴𝐵𝐶”〉:(0..^(#‘〈“𝐴𝐵𝐶”〉))⟶𝑃 ↔ 〈“𝐴𝐵𝐶”〉:{0, 1, 2}⟶𝑃) |
12 | 6, 11 | sylib 208 |
. . . 4
⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉:{0, 1, 2}⟶𝑃) |
13 | | fdm 6051 |
. . . 4
⊢
(〈“𝐴𝐵𝐶”〉:{0, 1, 2}⟶𝑃 → dom 〈“𝐴𝐵𝐶”〉 = {0, 1, 2}) |
14 | 12, 13 | syl 17 |
. . 3
⊢ (𝜑 → dom 〈“𝐴𝐵𝐶”〉 = {0, 1, 2}) |
15 | 14 | raleqdv 3144 |
. . 3
⊢ (𝜑 → (∀𝑗 ∈ dom 〈“𝐴𝐵𝐶”〉((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)))) |
16 | 14, 15 | raleqbidv 3152 |
. 2
⊢ (𝜑 → (∀𝑖 ∈ dom 〈“𝐴𝐵𝐶”〉∀𝑗 ∈ dom 〈“𝐴𝐵𝐶”〉((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)))) |
17 | | trgcgrg.p |
. . 3
⊢ 𝑃 = (Base‘𝐺) |
18 | | trgcgrg.m |
. . 3
⊢ − =
(dist‘𝐺) |
19 | | trgcgrg.r |
. . 3
⊢ ∼ =
(cgrG‘𝐺) |
20 | | trgcgrg.g |
. . 3
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
21 | | 0re 10040 |
. . . . 5
⊢ 0 ∈
ℝ |
22 | | 1re 10039 |
. . . . 5
⊢ 1 ∈
ℝ |
23 | | 2re 11090 |
. . . . 5
⊢ 2 ∈
ℝ |
24 | | tpssi 4369 |
. . . . 5
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ ∧ 2 ∈ ℝ) → {0, 1, 2}
⊆ ℝ) |
25 | 21, 22, 23, 24 | mp3an 1424 |
. . . 4
⊢ {0, 1, 2}
⊆ ℝ |
26 | 25 | a1i 11 |
. . 3
⊢ (𝜑 → {0, 1, 2} ⊆
ℝ) |
27 | | trgcgrg.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ 𝑃) |
28 | | trgcgrg.e |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ 𝑃) |
29 | | trgcgrg.f |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝑃) |
30 | 27, 28, 29 | s3cld 13617 |
. . . . 5
⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ Word 𝑃) |
31 | | wrdf 13310 |
. . . . 5
⊢
(〈“𝐷𝐸𝐹”〉 ∈ Word 𝑃 → 〈“𝐷𝐸𝐹”〉:(0..^(#‘〈“𝐷𝐸𝐹”〉))⟶𝑃) |
32 | 30, 31 | syl 17 |
. . . 4
⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉:(0..^(#‘〈“𝐷𝐸𝐹”〉))⟶𝑃) |
33 | | s3len 13639 |
. . . . . . 7
⊢
(#‘〈“𝐷𝐸𝐹”〉) = 3 |
34 | 33 | oveq2i 6661 |
. . . . . 6
⊢
(0..^(#‘〈“𝐷𝐸𝐹”〉)) = (0..^3) |
35 | 34, 9 | eqtri 2644 |
. . . . 5
⊢
(0..^(#‘〈“𝐷𝐸𝐹”〉)) = {0, 1, 2} |
36 | 35 | feq2i 6037 |
. . . 4
⊢
(〈“𝐷𝐸𝐹”〉:(0..^(#‘〈“𝐷𝐸𝐹”〉))⟶𝑃 ↔ 〈“𝐷𝐸𝐹”〉:{0, 1, 2}⟶𝑃) |
37 | 32, 36 | sylib 208 |
. . 3
⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉:{0, 1, 2}⟶𝑃) |
38 | 17, 18, 19, 20, 26, 12, 37 | iscgrgd 25408 |
. 2
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉 ↔ ∀𝑖 ∈ dom 〈“𝐴𝐵𝐶”〉∀𝑗 ∈ dom 〈“𝐴𝐵𝐶”〉((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)))) |
39 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 0 → (〈“𝐴𝐵𝐶”〉‘𝑗) = (〈“𝐴𝐵𝐶”〉‘0)) |
40 | | s3fv0 13636 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑃 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
41 | 1, 40 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
42 | 39, 41 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 0) → (〈“𝐴𝐵𝐶”〉‘𝑗) = 𝐴) |
43 | 42 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 0) → ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴)) |
44 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 0 → (〈“𝐷𝐸𝐹”〉‘𝑗) = (〈“𝐷𝐸𝐹”〉‘0)) |
45 | | s3fv0 13636 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑃 → (〈“𝐷𝐸𝐹”〉‘0) = 𝐷) |
46 | 27, 45 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐷𝐸𝐹”〉‘0) = 𝐷) |
47 | 44, 46 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 0) → (〈“𝐷𝐸𝐹”〉‘𝑗) = 𝐷) |
48 | 47 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 0) → ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷)) |
49 | 43, 48 | eqeq12d 2637 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 = 0) → (((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷))) |
50 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 1 → (〈“𝐴𝐵𝐶”〉‘𝑗) = (〈“𝐴𝐵𝐶”〉‘1)) |
51 | | s3fv1 13637 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ 𝑃 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
52 | 2, 51 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
53 | 50, 52 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 1) → (〈“𝐴𝐵𝐶”〉‘𝑗) = 𝐵) |
54 | 53 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 1) → ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵)) |
55 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 1 → (〈“𝐷𝐸𝐹”〉‘𝑗) = (〈“𝐷𝐸𝐹”〉‘1)) |
56 | | s3fv1 13637 |
. . . . . . . . . . 11
⊢ (𝐸 ∈ 𝑃 → (〈“𝐷𝐸𝐹”〉‘1) = 𝐸) |
57 | 28, 56 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐷𝐸𝐹”〉‘1) = 𝐸) |
58 | 55, 57 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 1) → (〈“𝐷𝐸𝐹”〉‘𝑗) = 𝐸) |
59 | 58 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 1) → ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸)) |
60 | 54, 59 | eqeq12d 2637 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 = 1) → (((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸))) |
61 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 2 → (〈“𝐴𝐵𝐶”〉‘𝑗) = (〈“𝐴𝐵𝐶”〉‘2)) |
62 | | s3fv2 13638 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ 𝑃 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
63 | 3, 62 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
64 | 61, 63 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 2) → (〈“𝐴𝐵𝐶”〉‘𝑗) = 𝐶) |
65 | 64 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 2) → ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶)) |
66 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 2 → (〈“𝐷𝐸𝐹”〉‘𝑗) = (〈“𝐷𝐸𝐹”〉‘2)) |
67 | | s3fv2 13638 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ 𝑃 → (〈“𝐷𝐸𝐹”〉‘2) = 𝐹) |
68 | 29, 67 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (〈“𝐷𝐸𝐹”〉‘2) = 𝐹) |
69 | 66, 68 | sylan9eqr 2678 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 = 2) → (〈“𝐷𝐸𝐹”〉‘𝑗) = 𝐹) |
70 | 69 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 = 2) → ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)) |
71 | 65, 70 | eqeq12d 2637 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 = 2) → (((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹))) |
72 | | 0red 10041 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℝ) |
73 | | 1red 10055 |
. . . . . . 7
⊢ (𝜑 → 1 ∈
ℝ) |
74 | 23 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℝ) |
75 | 49, 60, 71, 72, 73, 74 | raltpd 4315 |
. . . . . 6
⊢ (𝜑 → (∀𝑗 ∈ {0, 1, 2}
((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
76 | 75 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 0) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
77 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 0 → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘0)) |
78 | 77 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘0)) |
79 | 41 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
80 | 78, 79 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 0) → 𝐴 = (〈“𝐴𝐵𝐶”〉‘𝑖)) |
81 | 80 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐴 − 𝐴) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴)) |
82 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 0 → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘0)) |
83 | 82 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘0)) |
84 | 46 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“𝐷𝐸𝐹”〉‘0) = 𝐷) |
85 | 83, 84 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 0) → 𝐷 = (〈“𝐷𝐸𝐹”〉‘𝑖)) |
86 | 85 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐷 − 𝐷) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷)) |
87 | 81, 86 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝐴 − 𝐴) = (𝐷 − 𝐷) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷))) |
88 | 80 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐴 − 𝐵) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵)) |
89 | 85 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐷 − 𝐸) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸)) |
90 | 88, 89 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝐴 − 𝐵) = (𝐷 − 𝐸) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸))) |
91 | 80 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐴 − 𝐶) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶)) |
92 | 85 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 0) → (𝐷 − 𝐹) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)) |
93 | 91, 92 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝐴 − 𝐶) = (𝐷 − 𝐹) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹))) |
94 | 87, 90, 93 | 3anbi123d 1399 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 0) → (((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐴 − 𝐶) = (𝐷 − 𝐹)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
95 | 76, 94 | bitr4d 271 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 = 0) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐴 − 𝐶) = (𝐷 − 𝐹)))) |
96 | 75 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 1) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
97 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 1 → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘1)) |
98 | 97 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘1)) |
99 | 52 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
100 | 98, 99 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 1) → 𝐵 = (〈“𝐴𝐵𝐶”〉‘𝑖)) |
101 | 100 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐵 − 𝐴) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴)) |
102 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 1 → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘1)) |
103 | 102 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘1)) |
104 | 57 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“𝐷𝐸𝐹”〉‘1) = 𝐸) |
105 | 103, 104 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 1) → 𝐸 = (〈“𝐷𝐸𝐹”〉‘𝑖)) |
106 | 105 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐸 − 𝐷) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷)) |
107 | 101, 106 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 1) → ((𝐵 − 𝐴) = (𝐸 − 𝐷) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷))) |
108 | 100 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐵 − 𝐵) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵)) |
109 | 105 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐸 − 𝐸) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸)) |
110 | 108, 109 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 1) → ((𝐵 − 𝐵) = (𝐸 − 𝐸) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸))) |
111 | 100 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐵 − 𝐶) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶)) |
112 | 105 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 1) → (𝐸 − 𝐹) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)) |
113 | 111, 112 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 1) → ((𝐵 − 𝐶) = (𝐸 − 𝐹) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹))) |
114 | 107, 110,
113 | 3anbi123d 1399 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 1) → (((𝐵 − 𝐴) = (𝐸 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
115 | 96, 114 | bitr4d 271 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 = 1) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((𝐵 − 𝐴) = (𝐸 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)))) |
116 | 75 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 2) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
117 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 2 → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘2)) |
118 | 117 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 2) → (〈“𝐴𝐵𝐶”〉‘𝑖) = (〈“𝐴𝐵𝐶”〉‘2)) |
119 | 63 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 2) → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
120 | 118, 119 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 2) → 𝐶 = (〈“𝐴𝐵𝐶”〉‘𝑖)) |
121 | 120 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐶 − 𝐴) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴)) |
122 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑖 = 2 → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘2)) |
123 | 122 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 2) → (〈“𝐷𝐸𝐹”〉‘𝑖) = (〈“𝐷𝐸𝐹”〉‘2)) |
124 | 68 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 = 2) → (〈“𝐷𝐸𝐹”〉‘2) = 𝐹) |
125 | 123, 124 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 = 2) → 𝐹 = (〈“𝐷𝐸𝐹”〉‘𝑖)) |
126 | 125 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐹 − 𝐷) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷)) |
127 | 121, 126 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 2) → ((𝐶 − 𝐴) = (𝐹 − 𝐷) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷))) |
128 | 120 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐶 − 𝐵) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵)) |
129 | 125 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐹 − 𝐸) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸)) |
130 | 128, 129 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 2) → ((𝐶 − 𝐵) = (𝐹 − 𝐸) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸))) |
131 | 120 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐶 − 𝐶) = ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶)) |
132 | 125 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 = 2) → (𝐹 − 𝐹) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)) |
133 | 131, 132 | eqeq12d 2637 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 = 2) → ((𝐶 − 𝐶) = (𝐹 − 𝐹) ↔ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹))) |
134 | 127, 130,
133 | 3anbi123d 1399 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 = 2) → (((𝐶 − 𝐴) = (𝐹 − 𝐷) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹)) ↔ (((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐴) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐷) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐵) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐸) ∧ ((〈“𝐴𝐵𝐶”〉‘𝑖) − 𝐶) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − 𝐹)))) |
135 | 116, 134 | bitr4d 271 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 = 2) → (∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ ((𝐶 − 𝐴) = (𝐹 − 𝐷) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹)))) |
136 | 95, 115, 135, 72, 73, 74 | raltpd 4315 |
. . 3
⊢ (𝜑 → (∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2}
((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)) ↔ (((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐴 − 𝐶) = (𝐷 − 𝐹)) ∧ ((𝐵 − 𝐴) = (𝐸 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) ∧ ((𝐶 − 𝐴) = (𝐹 − 𝐷) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹))))) |
137 | | an33rean 1446 |
. . . 4
⊢ ((((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐴 − 𝐶) = (𝐷 − 𝐹)) ∧ ((𝐵 − 𝐴) = (𝐸 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) ∧ ((𝐶 − 𝐴) = (𝐹 − 𝐷) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹))) ↔ (((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹)) ∧ (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))))) |
138 | | eqid 2622 |
. . . . . . . 8
⊢
(Itv‘𝐺) =
(Itv‘𝐺) |
139 | 17, 18, 138, 20, 1, 27 | tgcgrtriv 25379 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝐴) = (𝐷 − 𝐷)) |
140 | 17, 18, 138, 20, 2, 28 | tgcgrtriv 25379 |
. . . . . . 7
⊢ (𝜑 → (𝐵 − 𝐵) = (𝐸 − 𝐸)) |
141 | 17, 18, 138, 20, 3, 29 | tgcgrtriv 25379 |
. . . . . . 7
⊢ (𝜑 → (𝐶 − 𝐶) = (𝐹 − 𝐹)) |
142 | 139, 140,
141 | 3jca 1242 |
. . . . . 6
⊢ (𝜑 → ((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹))) |
143 | 142 | biantrurd 529 |
. . . . 5
⊢ (𝜑 → ((((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))) ↔ (((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹)) ∧ (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))))) |
144 | | simprl 794 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷))) → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
145 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
146 | 20 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → 𝐺 ∈ TarskiG) |
147 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → 𝐴 ∈ 𝑃) |
148 | 2 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → 𝐵 ∈ 𝑃) |
149 | 27 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → 𝐷 ∈ 𝑃) |
150 | 28 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → 𝐸 ∈ 𝑃) |
151 | 17, 18, 138, 146, 147, 148, 149, 150, 145 | tgcgrcomlr 25375 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → (𝐵 − 𝐴) = (𝐸 − 𝐷)) |
152 | 145, 151 | jca 554 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸)) → ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷))) |
153 | 144, 152 | impbida 877 |
. . . . . 6
⊢ (𝜑 → (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐸))) |
154 | | simprl 794 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸))) → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
155 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
156 | 20 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → 𝐺 ∈ TarskiG) |
157 | 2 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → 𝐵 ∈ 𝑃) |
158 | 3 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → 𝐶 ∈ 𝑃) |
159 | 28 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → 𝐸 ∈ 𝑃) |
160 | 29 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → 𝐹 ∈ 𝑃) |
161 | 17, 18, 138, 156, 157, 158, 159, 160, 155 | tgcgrcomlr 25375 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → (𝐶 − 𝐵) = (𝐹 − 𝐸)) |
162 | 155, 161 | jca 554 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) → ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸))) |
163 | 154, 162 | impbida 877 |
. . . . . 6
⊢ (𝜑 → (((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ↔ (𝐵 − 𝐶) = (𝐸 − 𝐹))) |
164 | | simprr 796 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))) → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
165 | 20 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → 𝐺 ∈ TarskiG) |
166 | 3 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → 𝐶 ∈ 𝑃) |
167 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → 𝐴 ∈ 𝑃) |
168 | 29 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → 𝐹 ∈ 𝑃) |
169 | 27 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → 𝐷 ∈ 𝑃) |
170 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
171 | 17, 18, 138, 165, 166, 167, 168, 169, 170 | tgcgrcomlr 25375 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
172 | 171, 170 | jca 554 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) → ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))) |
173 | 164, 172 | impbida 877 |
. . . . . 6
⊢ (𝜑 → (((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) ↔ (𝐶 − 𝐴) = (𝐹 − 𝐷))) |
174 | 153, 163,
173 | 3anbi123d 1399 |
. . . . 5
⊢ (𝜑 → ((((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))) ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) |
175 | 143, 174 | bitr3d 270 |
. . . 4
⊢ (𝜑 → ((((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹)) ∧ (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐴) = (𝐸 − 𝐷)) ∧ ((𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸)) ∧ ((𝐴 − 𝐶) = (𝐷 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) |
176 | 137, 175 | syl5bb 272 |
. . 3
⊢ (𝜑 → ((((𝐴 − 𝐴) = (𝐷 − 𝐷) ∧ (𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐴 − 𝐶) = (𝐷 − 𝐹)) ∧ ((𝐵 − 𝐴) = (𝐸 − 𝐷) ∧ (𝐵 − 𝐵) = (𝐸 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹)) ∧ ((𝐶 − 𝐴) = (𝐹 − 𝐷) ∧ (𝐶 − 𝐵) = (𝐹 − 𝐸) ∧ (𝐶 − 𝐶) = (𝐹 − 𝐹))) ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) |
177 | 136, 176 | bitr2d 269 |
. 2
⊢ (𝜑 → (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) ↔ ∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2} ((〈“𝐴𝐵𝐶”〉‘𝑖) − (〈“𝐴𝐵𝐶”〉‘𝑗)) = ((〈“𝐷𝐸𝐹”〉‘𝑖) − (〈“𝐷𝐸𝐹”〉‘𝑗)))) |
178 | 16, 38, 177 | 3bitr4d 300 |
1
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉 ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) |