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Theorem trgcgrg 25128
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 13413 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5 wrdf 13111 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
64, 5syl 17 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
7 s3len 13435 . . . . . . . 8 (#‘⟨“𝐴𝐵𝐶”⟩) = 3
87oveq2i 6538 . . . . . . 7 (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
9 fzo0to3tp 12376 . . . . . . 7 (0..^3) = {0, 1, 2}
108, 9eqtri 2631 . . . . . 6 (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2}
1110feq2i 5936 . . . . 5 (⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
126, 11sylib 206 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
13 fdm 5950 . . . 4 (⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1412, 13syl 17 . . 3 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1514raleqdv 3120 . . 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 3128 . 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 9896 . . . . 5 0 ∈ ℝ
22 1re 9895 . . . . 5 1 ∈ ℝ
23 2re 10937 . . . . 5 2 ∈ ℝ
24 tpssi 4304 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 2 ∈ ℝ) → {0, 1, 2} ⊆ ℝ)
2521, 22, 23, 24mp3an 1415 . . . 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 13413 . . . . 5 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
31 wrdf 13111 . . . . 5 (⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 → ⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
3230, 31syl 17 . . . 4 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
33 s3len 13435 . . . . . . 7 (#‘⟨“𝐷𝐸𝐹”⟩) = 3
3433oveq2i 6538 . . . . . 6 (0..^(#‘⟨“𝐷𝐸𝐹”⟩)) = (0..^3)
3534, 9eqtri 2631 . . . . 5 (0..^(#‘⟨“𝐷𝐸𝐹”⟩)) = {0, 1, 2}
3635feq2i 5936 . . . 4 (⟨“𝐷𝐸𝐹”⟩:(0..^(#‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃 ↔ ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3732, 36sylib 206 . . 3 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3817, 18, 19, 20, 26, 12, 37iscgrgd 25126 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
39 fveq2 6088 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘0))
40 s3fv0 13432 . . . . . . . . . . 11 (𝐴𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
411, 40syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
4239, 41sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐴)
4342oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
44 fveq2 6088 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘0))
45 s3fv0 13432 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4627, 45syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4744, 46sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐷)
4847oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
4943, 48eqeq12d 2624 . . . . . . 7 ((𝜑𝑗 = 0) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
50 fveq2 6088 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘1))
51 s3fv1 13433 . . . . . . . . . . 11 (𝐵𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
522, 51syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
5350, 52sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐵)
5453oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
55 fveq2 6088 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘1))
56 s3fv1 13433 . . . . . . . . . . 11 (𝐸𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5728, 56syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5855, 57sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐸)
5958oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
6054, 59eqeq12d 2624 . . . . . . 7 ((𝜑𝑗 = 1) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
61 fveq2 6088 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘2))
62 s3fv2 13434 . . . . . . . . . . 11 (𝐶𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
633, 62syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
6461, 63sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐶)
6564oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
66 fveq2 6088 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘2))
67 s3fv2 13434 . . . . . . . . . . 11 (𝐹𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6829, 67syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6966, 68sylan9eqr 2665 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐹)
7069oveq2d 6543 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
7165, 70eqeq12d 2624 . . . . . . 7 ((𝜑𝑗 = 2) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
72 0red 9897 . . . . . . 7 (𝜑 → 0 ∈ ℝ)
73 1red 9911 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
7423a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
7549, 60, 71, 72, 73, 74raltpd 4257 . . . . . 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 479 . . . . 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 6088 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7877adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7941adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
8078, 79eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
8180oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
82 fveq2 6088 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8382adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8446adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
8583, 84eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐷 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
8685oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
8781, 86eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐴) = (𝐷 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
8880oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
8985oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
9088, 89eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐵) = (𝐷 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
9180oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
9285oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
9391, 92eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐶) = (𝐷 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
9487, 90, 933anbi123d 1390 . . . . 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 269 . . . 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 479 . . . . 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 6088 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9897adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9952adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
10098, 99eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
101100oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
102 fveq2 6088 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
103102adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
10457adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
105103, 104eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐸 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
106105oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
107101, 106eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐴) = (𝐸 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
108100oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
109105oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
110108, 109eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐵) = (𝐸 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
111100oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
112105oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
113111, 112eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐶) = (𝐸 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
114107, 110, 1133anbi123d 1390 . . . . 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 269 . . . 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 479 . . . . 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 6088 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
118117adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
11963adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
120118, 119eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
121120oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
122 fveq2 6088 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
123122adantl 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
12468adantr 479 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
125123, 124eqtr2d 2644 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐹 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
126125oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
127121, 126eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐴) = (𝐹 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
128120oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
129125oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
130128, 129eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐵) = (𝐹 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
131120oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
132125oveq1d 6542 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
133131, 132eqeq12d 2624 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐶) = (𝐹 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
134127, 130, 1333anbi123d 1390 . . . . 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 269 . . . 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 4257 . . 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 1437 . . . 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 2609 . . . . . . . 8 (Itv‘𝐺) = (Itv‘𝐺)
13917, 18, 138, 20, 1, 27tgcgrtriv 25096 . . . . . . 7 (𝜑 → (𝐴 𝐴) = (𝐷 𝐷))
14017, 18, 138, 20, 2, 28tgcgrtriv 25096 . . . . . . 7 (𝜑 → (𝐵 𝐵) = (𝐸 𝐸))
14117, 18, 138, 20, 3, 29tgcgrtriv 25096 . . . . . . 7 (𝜑 → (𝐶 𝐶) = (𝐹 𝐹))
142139, 140, 1413jca 1234 . . . . . 6 (𝜑 → ((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)))
143142biantrurd 527 . . . . 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 789 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷))) → (𝐴 𝐵) = (𝐷 𝐸))
145 simpr 475 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐴 𝐵) = (𝐷 𝐸))
14620adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐺 ∈ TarskiG)
1471adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐴𝑃)
1482adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐵𝑃)
14927adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐷𝑃)
15028adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐸𝑃)
15117, 18, 138, 146, 147, 148, 149, 150, 145tgcgrcomlr 25092 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐵 𝐴) = (𝐸 𝐷))
152145, 151jca 552 . . . . . . 7 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)))
153144, 152impbida 872 . . . . . 6 (𝜑 → (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ↔ (𝐴 𝐵) = (𝐷 𝐸)))
154 simprl 789 . . . . . . 7 ((𝜑 ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸))) → (𝐵 𝐶) = (𝐸 𝐹))
155 simpr 475 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐵 𝐶) = (𝐸 𝐹))
15620adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐺 ∈ TarskiG)
1572adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐵𝑃)
1583adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐶𝑃)
15928adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐸𝑃)
16029adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐹𝑃)
16117, 18, 138, 156, 157, 158, 159, 160, 155tgcgrcomlr 25092 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐶 𝐵) = (𝐹 𝐸))
162155, 161jca 552 . . . . . . 7 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)))
163154, 162impbida 872 . . . . . 6 (𝜑 → (((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ↔ (𝐵 𝐶) = (𝐸 𝐹)))
164 simprr 791 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) → (𝐶 𝐴) = (𝐹 𝐷))
16520adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐺 ∈ TarskiG)
1663adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐶𝑃)
1671adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐴𝑃)
16829adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐹𝑃)
16927adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐷𝑃)
170 simpr 475 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐶 𝐴) = (𝐹 𝐷))
17117, 18, 138, 165, 166, 167, 168, 169, 170tgcgrcomlr 25092 . . . . . . . 8 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐴 𝐶) = (𝐷 𝐹))
172171, 170jca 552 . . . . . . 7 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))
173164, 172impbida 872 . . . . . 6 (𝜑 → (((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ↔ (𝐶 𝐴) = (𝐹 𝐷)))
174153, 163, 1733anbi123d 1390 . . . . 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 268 . . . 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 270 . . 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 267 . 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 298 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wss 3539  {ctp 4128   class class class wbr 4577  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  1c1 9793  2c2 10917  3c3 10918  ..^cfzo 12289  #chash 12934  Word cword 13092  ⟨“cs3 13384  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  cgrGccgrg 25123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgcb 25066  df-trkg 25069  df-cgrg 25124
This theorem is referenced by:  trgcgr  25129  cgr3simp1  25133  cgr3simp2  25134  cgr3simp3  25135  cgraswap  25430
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