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Theorem trgcgrg 25455
 Description: The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
trgcgrg.p 𝑃 = (Base‘𝐺)
trgcgrg.m = (dist‘𝐺)
trgcgrg.r = (cgrG‘𝐺)
trgcgrg.g (𝜑𝐺 ∈ TarskiG)
trgcgrg.a (𝜑𝐴𝑃)
trgcgrg.b (𝜑𝐵𝑃)
trgcgrg.c (𝜑𝐶𝑃)
trgcgrg.d (𝜑𝐷𝑃)
trgcgrg.e (𝜑𝐸𝑃)
trgcgrg.f (𝜑𝐹𝑃)
Assertion
Ref Expression
trgcgrg (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))

Proof of Theorem trgcgrg
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trgcgrg.a . . . . . . 7 (𝜑𝐴𝑃)
2 trgcgrg.b . . . . . . 7 (𝜑𝐵𝑃)
3 trgcgrg.c . . . . . . 7 (𝜑𝐶𝑃)
41, 2, 3s3cld 13663 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5 wrdf 13342 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
64, 5syl 17 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
7 s3len 13685 . . . . . . . 8 (#‘⟨“𝐴𝐵𝐶”⟩) = 3
87oveq2i 6701 . . . . . . 7 (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
9 fzo0to3tp 12594 . . . . . . 7 (0..^3) = {0, 1, 2}
108, 9eqtri 2673 . . . . . 6 (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2}
1110feq2i 6075 . . . . 5 (⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
126, 11sylib 208 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
13 fdm 6089 . . . 4 (⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1412, 13syl 17 . . 3 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1514raleqdv 3174 . . 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))))
1614, 15raleqbidv 3182 . 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 (𝜑𝐺 ∈ TarskiG)
21 0re 10078 . . . . 5 0 ∈ ℝ
22 1re 10077 . . . . 5 1 ∈ ℝ
23 2re 11128 . . . . 5 2 ∈ ℝ
24 tpssi 4401 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 2 ∈ ℝ) → {0, 1, 2} ⊆ ℝ)
2521, 22, 23, 24mp3an 1464 . . . 4 {0, 1, 2} ⊆ ℝ
2625a1i 11 . . 3 (𝜑 → {0, 1, 2} ⊆ ℝ)
27 trgcgrg.d . . . . . 6 (𝜑𝐷𝑃)
28 trgcgrg.e . . . . . 6 (𝜑𝐸𝑃)
29 trgcgrg.f . . . . . 6 (𝜑𝐹𝑃)
3027, 28, 29s3cld 13663 . . . . 5 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
31 wrdf 13342 . . . . 5 (⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 → ⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
3230, 31syl 17 . . . 4 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
33 s3len 13685 . . . . . . 7 (#‘⟨“𝐷𝐸𝐹”⟩) = 3
3433oveq2i 6701 . . . . . 6 (0..^(#‘⟨“𝐷𝐸𝐹”⟩)) = (0..^3)
3534, 9eqtri 2673 . . . . 5 (0..^(#‘⟨“𝐷𝐸𝐹”⟩)) = {0, 1, 2}
3635feq2i 6075 . . . 4 (⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃 ↔ ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3732, 36sylib 208 . . 3 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3817, 18, 19, 20, 26, 12, 37iscgrgd 25453 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
39 fveq2 6229 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘0))
40 s3fv0 13682 . . . . . . . . . . 11 (𝐴𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
411, 40syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
4239, 41sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐴)
4342oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
44 fveq2 6229 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘0))
45 s3fv0 13682 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4627, 45syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4744, 46sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐷)
4847oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
4943, 48eqeq12d 2666 . . . . . . 7 ((𝜑𝑗 = 0) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
50 fveq2 6229 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘1))
51 s3fv1 13683 . . . . . . . . . . 11 (𝐵𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
522, 51syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
5350, 52sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐵)
5453oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
55 fveq2 6229 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘1))
56 s3fv1 13683 . . . . . . . . . . 11 (𝐸𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5728, 56syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5855, 57sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐸)
5958oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
6054, 59eqeq12d 2666 . . . . . . 7 ((𝜑𝑗 = 1) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
61 fveq2 6229 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘2))
62 s3fv2 13684 . . . . . . . . . . 11 (𝐶𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
633, 62syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
6461, 63sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐶)
6564oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
66 fveq2 6229 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘2))
67 s3fv2 13684 . . . . . . . . . . 11 (𝐹𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6829, 67syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6966, 68sylan9eqr 2707 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐹)
7069oveq2d 6706 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
7165, 70eqeq12d 2666 . . . . . . 7 ((𝜑𝑗 = 2) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
72 0red 10079 . . . . . . 7 (𝜑 → 0 ∈ ℝ)
73 1red 10093 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
7423a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
7549, 60, 71, 72, 73, 74raltpd 4346 . . . . . 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))))
7675adantr 480 . . . . 5 ((𝜑𝑖 = 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 6229 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7877adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7941adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
8078, 79eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
8180oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
82 fveq2 6229 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8382adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8446adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
8583, 84eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐷 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
8685oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
8781, 86eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐴) = (𝐷 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
8880oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
8985oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
9088, 89eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐵) = (𝐷 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
9180oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
9285oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
9391, 92eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐶) = (𝐷 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
9487, 90, 933anbi123d 1439 . . . . 5 ((𝜑𝑖 = 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))))
9576, 94bitr4d 271 . . . 4 ((𝜑𝑖 = 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))))
9675adantr 480 . . . . 5 ((𝜑𝑖 = 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 6229 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9897adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9952adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
10098, 99eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
101100oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
102 fveq2 6229 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
103102adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
10457adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
105103, 104eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐸 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
106105oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
107101, 106eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐴) = (𝐸 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
108100oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
109105oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
110108, 109eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐵) = (𝐸 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
111100oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
112105oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
113111, 112eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐶) = (𝐸 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
114107, 110, 1133anbi123d 1439 . . . . 5 ((𝜑𝑖 = 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))))
11596, 114bitr4d 271 . . . 4 ((𝜑𝑖 = 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))))
11675adantr 480 . . . . 5 ((𝜑𝑖 = 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 6229 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
118117adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
11963adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
120118, 119eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
121120oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
122 fveq2 6229 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
123122adantl 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
12468adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
125123, 124eqtr2d 2686 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐹 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
126125oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
127121, 126eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐴) = (𝐹 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
128120oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
129125oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
130128, 129eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐵) = (𝐹 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
131120oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
132125oveq1d 6705 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
133131, 132eqeq12d 2666 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐶) = (𝐹 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
134127, 130, 1333anbi123d 1439 . . . . 5 ((𝜑𝑖 = 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))))
135116, 134bitr4d 271 . . . 4 ((𝜑𝑖 = 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))))
13695, 115, 135, 72, 73, 74raltpd 4346 . . 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 1486 . . . 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 2651 . . . . . . . 8 (Itv‘𝐺) = (Itv‘𝐺)
13917, 18, 138, 20, 1, 27tgcgrtriv 25424 . . . . . . 7 (𝜑 → (𝐴 𝐴) = (𝐷 𝐷))
14017, 18, 138, 20, 2, 28tgcgrtriv 25424 . . . . . . 7 (𝜑 → (𝐵 𝐵) = (𝐸 𝐸))
14117, 18, 138, 20, 3, 29tgcgrtriv 25424 . . . . . . 7 (𝜑 → (𝐶 𝐶) = (𝐹 𝐹))
142139, 140, 1413jca 1261 . . . . . 6 (𝜑 → ((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)))
143142biantrurd 528 . . . . 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 809 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷))) → (𝐴 𝐵) = (𝐷 𝐸))
145 simpr 476 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐴 𝐵) = (𝐷 𝐸))
14620adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐺 ∈ TarskiG)
1471adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐴𝑃)
1482adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐵𝑃)
14927adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐷𝑃)
15028adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐸𝑃)
15117, 18, 138, 146, 147, 148, 149, 150, 145tgcgrcomlr 25420 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐵 𝐴) = (𝐸 𝐷))
152145, 151jca 553 . . . . . . 7 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)))
153144, 152impbida 895 . . . . . 6 (𝜑 → (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ↔ (𝐴 𝐵) = (𝐷 𝐸)))
154 simprl 809 . . . . . . 7 ((𝜑 ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸))) → (𝐵 𝐶) = (𝐸 𝐹))
155 simpr 476 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐵 𝐶) = (𝐸 𝐹))
15620adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐺 ∈ TarskiG)
1572adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐵𝑃)
1583adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐶𝑃)
15928adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐸𝑃)
16029adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐹𝑃)
16117, 18, 138, 156, 157, 158, 159, 160, 155tgcgrcomlr 25420 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐶 𝐵) = (𝐹 𝐸))
162155, 161jca 553 . . . . . . 7 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)))
163154, 162impbida 895 . . . . . 6 (𝜑 → (((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ↔ (𝐵 𝐶) = (𝐸 𝐹)))
164 simprr 811 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) → (𝐶 𝐴) = (𝐹 𝐷))
16520adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐺 ∈ TarskiG)
1663adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐶𝑃)
1671adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐴𝑃)
16829adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐹𝑃)
16927adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐷𝑃)
170 simpr 476 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐶 𝐴) = (𝐹 𝐷))
17117, 18, 138, 165, 166, 167, 168, 169, 170tgcgrcomlr 25420 . . . . . . . 8 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐴 𝐶) = (𝐷 𝐹))
172171, 170jca 553 . . . . . . 7 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))
173164, 172impbida 895 . . . . . 6 (𝜑 → (((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ↔ (𝐶 𝐴) = (𝐹 𝐷)))
174153, 163, 1733anbi123d 1439 . . . . 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))))
175143, 174bitr3d 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))))
176137, 175syl5bb 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))))
177136, 176bitr2d 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))))
17816, 38, 1773bitr4d 300 1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  {ctp 4214   class class class wbr 4685  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975  2c2 11108  3c3 11109  ..^cfzo 12504  #chash 13157  Word cword 13323  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  cgrGccgrg 25450 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgcb 25394  df-trkg 25397  df-cgrg 25451 This theorem is referenced by:  trgcgr  25456  cgr3simp1  25460  cgr3simp2  25461  cgr3simp3  25462  cgraswap  25757
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